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» MATHEMATICS
PRACTICAL MEN:
BEnre
A COMMONPLACE BOOK
or
PURE AND MIXED MATHEMATICS,
DBSIOVSD OHHILT rOR THB VSE OF
CIVIL ENGINEERS, ARCHITECTS, AND SURVEYORS.
BY OLINTHUS GBEGOBY, LL.D., F.R.A.S.
THIRD EDITION, REVISED AND ENLARGED
BY HENRY LAW,
cnriL xiranriVB.
LONDON:
JOHN WEALE, 69, HIGH HOLBORN.
1848.
LONDON :
OBOROB WOODFALL AND SOX,
AJfOCL COURT. 8KINNBR STRBKT.
TU
SIR ISAMBART BRUNEL, F.RS.,
na BO. xra
THIS WORK.
DESIGNED FOB THE USE OF MEMBEBS OF THAT FBOFESSION WHICH
HE HAS 80 OBEATLY ADVANCED,
IS INSCBIBED,
, IN GBATEFUL ACKKOWLEDOMENT OF MANY KINDNESSES,
BY HIS OBUOED FBIEND AND PUPIL,
HENRY LAW.
AUT HOE'S PREFACE.
Ths wot\l now presented to the public bad its origin in a desire
which I felt to draw up an Essay on the principles and apph'cations
of the Tnechanical sciences for the use of the younger members of the
InsdtutioQ of Civil Engineers. The eminent individuals who are
deserredly r^arded as the main pillars of that useful Institution,
stand in need of no such instructions as are in my power to impart :
kt it seemed expedient to prepare an Essay, comprised within
moderate limits, which might furnish scientific instruction for the
many young men of ardour and enterprise who have of late years
deToted themseWes to the interesting and important profession, of
whose members that InsUtution is principally constituted. My first
design was to compose a paper which might be read at one or two
of the meetings of that Society ; but, as often happens in such cases,
the embryo thought has grown, during meditation, from an essay to a
book ; and what was first meant to be a very compendious selection
of principles and rules, has, in its execution, assumed the appearance
of a systematic aualysis of principles, theorems, rules, and tables.
Indeed, tbe circumstances in which the inhabitants of this country
are now placed, with regard to the Ioto and acquisition of knowledge
impelled me, almost unconscionsly, to such an extension of my ori
ginal plan, as sprung from a desire to contribute to the instruction of
that numerous class, the practical mechanics of this country. Besides
the early disadvantages under which many of them have laboured,
there is another which results from the activity of their pursuits.
Unable, therefore, to go through the details of an extensive systematie
course, they must, for the most part, be satisfied with imperfect views
of theories and principles, and take much upon trust : an evil, how
ever, which tbe establishment of Societies, and the composition of
treatises, with an express view to their benefit, will probably soon
diminish.
Lord Brouorjim, in his ^^ PracHeal Observations upon the Educa^
turn of the Peofk^ remarks that " a most essential service will be
▼1 AUTHORS PREFACE.
rendered to the cause of knowle<lge, by him who shall devote his
time to the composition of elementary treatises on the Mathematics,
sufficiently clear, and yet sufficiently compendious, to exemplify the
metliod of reasoning employed in that science, and to impart an
accurate knowledge of the most useful fundamental propositions,
with their application to practical purposes; and treatises upon Na
tural Philosophy, which may teach the great principles of physics,
and their practical application, to readers who have but a general
knowledge of Mathematics, or who are even wholly ignorant of the
science beyond the common rules of arithmetic/' And again, '^ He
who shall prepare a treatise simply and concisely imfolding the doc
trines of Algebra, Geometry, and Mechanics, and adding examples
calculated to strike the imagination, of their connection with other
branches of knowledge, and with the arts of common life, may fairly
claim a large share in that rich harvest of discovery and invention
which must be reaped by the thousands of ingenious and active men,
thus enabled to bend their faculties towards objects at once useful
and sublime."
I do not attempt to persuade myself that the present volume will
be thought adequately to supply the desiderata to which these pas
sages advert : yet I could not but be gratified, after full twothirds
of it were written, to find that the views which guided me in its
execution accorded so far with the judgment of an individual, dis
tinguished, as Lord Brougham was, in early life, for the elegance
and profundity of his mathematical researches.
With a view to the elementary instruction of those who have not
previously studied Mathematics, I have commenced with brief, but, I
hope, perspicuous, treatises on Arithmetic and Algebra; a competent
acquaintance with both of these being necessary to ensure that ac
curacy in computation which every practical man ought to attain,
and that ready comprehension of scientific theorems and formul»
which becomes the key to the stores of higher knowledge. As no
man sharpens his tool or his weapon, merely that it may be sharp,
but that it may be the fitter for use ; so no thoughtful man learns
Arithmetic and Algebra for the mere sake of knowing those branches
of science, but that he may employ them ; and these being possessed
as valuable prerequisites, the course of an author is thereby facili
tated : for then, while he endeavours to express even common mat
ters so that the learned shall not be disgusted, he may so express the
more abstract and difficult that the comparatively ignorant (and the
▲UTHOBS PRBPACI. yil
DereliQowledge of Jirithmetie tmdAl^bra is, in our times, compara^
the i^orence,') may practically understand and apply them.
Mter the first 97 I>age8, the remaining matter is synoptical. The
leoend topics of Oeomctry, Trigonometry, Conic Sections, Curves,
Pmpectiye, Mensuration, Statics, Dynamics, Hydrostatics, Hydro
dyiuumcs, and Pneumatics are thus treated. The definitions and
pnudples are exhibited in an orderly series ; but investigations and
demonstiations are only sparingly introduced. This portion of the
voik is aldn in its nature to a syllabus of a Course of Lectures on
tiie deportments of science which it treats; with this difference,
however, occasioned by tbe leading object of the publication, that
popular iUostrations are more frequently introduced, practical appli
eatioDs incessantly borne in mind, and such tables as seemed best
odeakied to save tbe labour of Architects, Mechanics, and Civil
EngineerB, inserted under their appropriate heads. Of these latter,
several have been collected from former treatises, &c., but not a few
have been either eompated or contributed expressly for this Common
place Book.
In a work like this, it would be absurd to pretend to originality.
Tbe plan, arrangement, and execution, are my own ; but the mate
rials have long been regarded, and rightly, as common property. It
baa been my aim to reduce them into the smallest possible space,
consiatratly with my general object ; but wherever I have found the
work in this respect prepared to my hands, I have transcribed it into
tbe following pages, with the usual references to the sources from
whence it was taken. They who are conversant with the best writers
on subjects of mixed nuitbematics and natural philosophy, will know
that Smeaion^ Bobison^ Play/air^ Young^ Du Buat^ Leslie^ Hachette^
Blandy Tredgold^ &c., are authors who ought to be consulted, n the
preparation of a volume like this. I hope it will appear that I have
duly, yet, at the same time, honourably, availed myself of the advan
tages which they supply. I have also made such selections from my
own earlier publications as were obviously suitable to my present
purpose ; but not so copiously, I trust, as to diminish the utility of
those volumes, or to make me an unfair borrower even from myself.
Besides our junior Civil Engineers, and the numerous Practical
Mechanics who are anxious to store their minds with scientific facts
and principles ; there are others to whom, I flatter myself, the fol
lowing pages will be found useful. Teachers of Mathematics, and
those departments of natural philosophy which are introduced into
TUl AUTHORS PRBFACB.
our more respectable seminaries, may probably find tbis volume to
occupy a convenient intermediate station between tbe merely popular
exbibitions of tbe trutbs of Mecbanics, Hydrostatics, &c., and tbe
larger treatises in wliicb tbe wliole cbain of inquiry and demonstra
tion is carefully presented link by link, and tbe successive portions
firmly connected upon irrefragable principles. Wbile students wbo
have recently terminated a scientific course, wbetber in our univer
sities, or otber institutions public or private, may, I would fain be
lieve, find in tbis Common place Book an abridged repository of tbe
most valuable principles and tbeorems, and of bints for tbeir applica
tions to practical purposes.
Tbe only performances witb wbicb I am acquainted, tbat bear any
direct analogy to tbis, are Martin's Young Students Memorial Bookj
Jones's Synopsis Pcdmariorum Matkeseos^ and Brunton's Compendium
of Mechanics; tbe latter of wbicb I had not seen until tbe present
volume was nearly completed. Tbe first and last mentioned of tbese
are neat and meritorious productions ; but restricted in tbeir utility
by tbe narrow space into wbicb tbey are compressed. Tbe otber,
written by the father of tbe late Sir William Jones, is a truly elegant
introduction to tbe principles of Mathematics, considering tbe time
in wbicb it was written (1706); but as it is altogether theoretical,
and is, moreover, now becoming exceedingly scarce, it by no means
supersedes tbe necessity^ for such I have been induced to regard it, of
a Compendium like tbat wbicb I now offer to the public.
In its execution I have aimed at no higher reputation than tbat of
being perspicuous, correct, and useful ; and if I shall be so fortunate
as to have succeeded in those points, I shall be perfectly satisfied.
Olinthus Gregory.
Boyal Military Academy,
Woolwich, October 1st, 1825.
In this new edition I have corrected a few errors wbicb had escaped
my notice in the former impression. I have also made a few such
additions and improvements as the lapse of time and tbe progress of
discovery rendered desirable ; and such as will, I hope, give the work
new claims on public approbation.
July let, 1883.
THE EDITOR'S PREFACE.
In presenting to the public a new edition of Dr. Gregory's " Mathe
matics for Practical Men,** the Editor feels that a few introductory
remarks are due from him to his readers. Not, indeed, to apologise
for presenting to them again a work which they have already
marked with their approbation, and which has from time to time
been favourably noticed by other writers, but to offer some explana
tion of the numerous alterations which he has taken upon himself to
make, and to state his reasons for having done so.
In looking through the former edition, it was evident, that since
the date of its publication, many of the subjects which it treated
upon had been greatly improved and extended ; railways had been
brought into successful operation ; the steam engine and machinery
generally had undergone vast improvements ; and almost every branch
of mechanical science had progressed in some degree ; it was there
fore found necessary, in order to render the present edition as exten
sively useful as its predecessors had been, that the work should
undergo an entire revision. In doing this, it was found that so large
a quantity of fresh matter would have to be given, as would swell
the Tolume to an inconvenient size, and by a necessary increase of
price, render the work less accessible to that important class — the
working mechanic9*for whose use it was so peculiarly adapted. To
obviate, therefore, these disadvantages, a smaller type has been
adopted, so that the work, although not much increased in its exter
nal dimensions, contains more than a third of its present bulk of new
matter.
In order to add to the usefulness of the work, and render it more
generally convenient as a book of reference, a more orderly arrange
ment of its contents has been adopted, and several articles, which,
although of much value, did not properly belong to the subject of the
work, have been thrown into an Appendix ; in addition to which a
Table of Contents has been given, showing at once the general ar
X KDIT0B8 PREPACB.
rangement and division of the work, as also a very copious Index at
the end.
The Editor has endeavoured, in the following short review of the
work in its present form, to point out the principal alterations or
additions which have heen made : —
The work has been divided into two distinct parts ; viz. 1st, Purb
Mathematics, comprising chapters upon Arithmetic, Algebra, Geo
metry, Mensuration, Trigonometry, Conic Sections, and the Proper
ties of Curves; and, 2ndly, Mixed or Applied Mathematics, being
the application of Mathematics to the general pursuits of the Engineer,
the Mechanician, the Surveyor, &c., comprised in chapters upon
Mechanics generally. Statics, Dynamics, Hydrostatics, Hydrody
namics, Pneumatics, Mechanical Agents, or Moving Powers, and the
Strength of Materials.
In the part upon Pure Mathematics : —
In the First Chapter, upon Arithmetic, the article npon fractions
has been extended, and rules given for the reduction of circulating
decimals; and an entirely new section (IX.) is added upon Loga
rithmic Arithmetic, containing a description of the Logarithmic
Tables given in the Appendix, with the method of using them,
and roles for performing the ordinary processes of calculation by
tlieir aid.
The Second Chapter, on Algebra, has undergone a very careful re
vision. In Section V., on Involution, a table of roots and powers of
monomials has been added ; and in the following section on Evolu
tion, a new rule for finding the roots of powers in general. In
Section X. examples have been added of the method for determining
the roots of equations. Section XII., on Fractional and Negative
Exponents, is entirely new ; as is also nearly the whole of the next
section on Logarithms ; to which have been added tables for converting
Common Logarithms into Hyperbolic^ and vice versd.
The Third Chapter, on Geometry, may almost be said to have been
rewritten. The definitions and propositions have been arranged in a
more orderly manner, and many additional ones have been added.
No demonstration of the several propositions will be found, as the
size of the work would not allow, nor did its practical character re
quire, that such should be given; but for the convenience of those
who may wish to see them demonstrated, a reference has been made to
BDIT0B8 PEBFACB. Xl
Euclid, giving the number of the theorem and book in which the
demonstration of the same proposition will be there found.
The Fourth Chapter, on Mensuration, has also been almost entireiy
rewritten : the tables of weights and measures have been put under a
more conYenient form, and rules and formuIfiB giveft for the mensura
tion of all kinds of superficies and solids.
In the Fifth Chapter, on Trigonometry, yery exteneiTe 2Vt^ofio
metrieal Formulm have \eea added, which cannot fail to be found of
serrioe in the pursuits of Engineers and Surveyors. The second sec
tion is entirely new, being a description of the Tables of Logarithmic
sines, cosines, tangents, cotangents, secants, and cosecants given in
the Appendix, with the method of using them.
The Sixth Chapter, on Conic Sections, has been almost entirely
rewritten, and several additional formuls and problems relating to
them have been added.
The Seventh Chapter, on the Properties of Curves, has also been
much extended ; the first section of definitions, and the latter part of
the fourth section, on the Epicycloid, being entirely new.
In the second division of the work, upon Mixsd Mathbm atigs:—
In the First Chapter, on Mechanics in General, many fresh defi
nitions have been added.
In the Second Chapter, the whole of the definitions and formulfs
in the first section, on Statical Equilibrium, are new. In the second
section, the principle of parallel presmret is applied to determine
the position of the center of gravity, and an example is given of the
centrobaryc method of determining the superficies or solidity of
surfaces or solids. In the third section, on the application of Statics
to the equilibrium of structures, the formuln have been more conve
niently arranged, and tables have been added of the natural slope
assumed by various kinds of materials, as also of the limiting angle
and coefficient of friction of the various materials used in the con
struction of arches, &c, the latter extracted from Professor Moseley's
work on the Mechanical Principles of Engineering. An article is
also added upon Suspension Bridges, a subject of much importance
to Engineers, from the general use into which they are being brought.
Formultt are given for determining all their elements and propor
tions, the use of which is illustrated by an example. In the Third
Chapter, on Dynamics, the definitions in the first section have been
nuidi ezlended ; and at the commencement of the second section, on
XU EDITORS PREFACE.
the General Laws of Uniform and Variable Motion, it has been en
deayoured to put in a clear light the long disputed question relative
to the momentum and vis viva of moving bodies. In this section,
also, formulse have been added expressing the relation between the
times, velocities, spaces, &c. of bodies in motion. In the third sec
tion some practical formulce are added for determining the proportions
for the coniccd governor^ so generally used for regulating the velocity
of steam engines and water wheels. In the fifth section, on the
Mechanical Powers, their true nature is explained, in order to pre
yent any misconception which might be occasioned by their designa
tion ; and a table is added exhibiting the ratio in each, between the
power exerted and the effect produced.
In the Fourth Chapter, on Hydrostatics, the fifth section, on Capil
lary Attraction, is entirely new.
In the Fifth Chapter, on Hydrodynamics, several additional for
malse are given, in the second section, for determining the velocity
and discharge through conduit pipes ; and the whole of the formulse
have been arranged in a more orderly manner, and numbered for the
convenience of reference.
In the Sixth Chapter, on Pneumatics, at the conclusion of the
first section, a very exact formul«B of Mr. Galbraith's is given, for
determining the velocity of sound, and for measuring distances by the
same.
In the Seventh Chapter, on the Mechanical Agents, the third sec
tion, on the Steam Engine, is entirely new. The nature and pro
perties of steam are explained, and formulse and rules are given, and
illustrated by examples, for determining its pressure or temperature.
The subject of the specific heat of steam is also explained, and for
mulse given for determiHing it from the pressure; and a table is
added of the pressure, temperature, and specific volume of steam,
from 5 lbs. to 1 50 lbs. upon the square inch. A general description
of the mode of action of the steam engine is given, which is fol
lowed by an investigation of its general theory upon the principles
first laid down by the Comte de Pambour. The various kinds of
engines employed are then classified, and described separately ; the
principles previously educed are applied to each ; and formulse are
deduced for the velocity of the engine, the quantity of steam used,
and the useful effect of the engine, for each particular case, which
are given both in a general and a more practical form. In this por
tion of the work, in treating of engines acting expansively, it was
EDITORS PREFACE. Xli)
found imposnble to dispense with the use of the Differential Calculus,
the introduction of which had heen studiously avoided, in order that
the work might be accesable to those who had not entered upon the
higher branches of Mathematics. The use of the Differential Cal
culus, howeyer, in the present case, while it was necessary for the
proper investigation of the subject, will not, it is hoped, in any way
abridge the usefulness of the work to those who are not conversant
wiUi its principles, since the results obtained are expressed in the
ordinary form, while for those who do understand them, it was
thought that it would prove much more satisfJEictory to them to be
enabled to examine and satisfy themselves of the truthfulness of the
several investigations.
The Eighth Chapter, on the Strength of Materials, has been
almost entirely rewritten. This portion of the work has been
rendered much more valuable by the introduction of the results of
Professor Hodgkinson and Mr. Cubitt's researches upon this subject,
«o important to the Civil Engineer and Architect.
The woodcuts have been replaced by copperplate engravings, and
several additional plates have been given, for three of which the Editor
is indebted to Mr. R. A. Rumble.
An Appendix has been added, containing a Table of Logarithmic
Differences, in a new, and, it is hoped, more convenient form than
that usually adopted, and Tables of the Logarithms of Numbers, and
of Logarithmic sines, tangents, &c. : these tables will be found of
great utility to Engineers and others, much engaged in calculations.
It also contains a new Table ( VL) of various useful numbers, with
their logarithms, and a Table (XI.) of the weight of materials fre
ijuently employed in construction. The remaining portion of the
Appendix consists of matter which stood in the body of the former
edition of the work, but which has now been put into the Appendix,
because, although too valuable to be omitted, it did not properly be
long to, but was only incidentally connected with, the subjects
therein treated.
The Editor has thought it t>nly doing justice to the late Dr. Ore
gory to point out thus in detail the extent of the alterations which he
has made, in order that the public may know how far each is re
sponsible for the work in its present form, and to prevent either
blame being attached to Dr. Gregory, or credit given to the Editor,
where both might have been misplaced.
H. L.
London,
21ft October, 1847.
CONTENTS.
PART I.— PUKE MATHEMATICS.
CHAPTER I.
Arithmetic.
PACK
Sect. 1. Definitions and Notation. 1
2. Addition of Whole Numben 5
S. Subtraction of Whole Numben 6
4. Multiplication of Whole Numben 7
5. DiTinon of Whole Numben 10
Proof of the fint Four Rules of Arithmetic . . . .Id
6. Vulgar Fractions 18
Reduction of Vulgar Fractions . . . . . .14
Addition and Subtraction of Vulgar FVactions . .17
Multiplication and Division of Vulgar Fractions . . 17
7. Decimal Fractions 18
Reduction of Decimals 19
Addition and Subtraction of Decimals 22
Multiplication and Division of Decimals 22
8. Complex Fractions used in the Arts and Commerce . .23
Reduction 28
Addition 24
Subtraction and Multiplication 25
Division * . .26
Duodecimals 27
9. Powen and Roots 28
Evolution 29
10. Ph)portion 82
Rule of Three 84
Determination of Ratios 87
11. Logarithmic Arithmetic ^
Use of die Tablet 42
Multiplication and Division bv Logarithms . . • . ^
Proportion, or the Rule of l^ree, by Logarithms . . 46
Evolution and Involution by Lo^urithms .47
12. Properties of Numben 49
CHAPTER IL
Algbbba.
8acr. I. Definitions and NoCatioti A2
2. AdfidoD and Subtraction ^
XVI CONTENTS.
PAGE
Sect. S. Multiplication 56
4. Division 58
5. Involution 60
6. Evolution 63
7. Surds 65
Reduction 65
Addition, Subtraction, and Multiplication .... 68
Division, Involution, and Evolution 69
8. Simple Equations 70
Extermination 73
Solution of Genera] Problems 75
9. Quadratic Equations 77
10. Equations in General 81
11. Progression 84
Arithmetical Progression 85
Geometrical Progression 86
12. Fractional and Negative Exponents 88
13. Logarithms 90
14. Computation of Formulae 95
CHAPTER in.
Geometry.
Sect. 1. Definitions 98
2. Of Angles, and Right Lines^ and their Rectangles . . .99
a Of Triangles 100
4. Of Quadrilaterals and Polygons 102
5 Of the Circle, and Inscribed and Circumscribed Figures . .104
6. Of Planes and Solids 109
7. Practical Geometry 112
CHAPTER IV.
Men&uration.
Sect. 1. Weights and Measures 119
1. Measures of Length 120
2. Measures of Sur&e 12]
3. Measures of Solidity and Capacity 121
4. Measures of Weight 123
5. Angular Measure 124
6. Measure of Time 124
Comparison of English and French Weights and Measures . 125
2. Mensuration of Supeirficies ] 27
3. Mensuration of Solids 130
CHAPTER V,
Trigonometry.
Sect. 1. Definitions and Triffonometrical Formulae 134
2. Trigonometrical Tables 139
CONTENTS. XVU
PAGE
SxcT. Sw Geoenl Propoeitkms 146
4. Solution of the Casei of Plane Triangles 148
Rifffatangled Plane Triangles 151
5. On Uie application of Trigonometry to Measuring Heights and
Distances 152
Detenninatipn of Heights and Distances by Approximate Me
chanical Methods 156
CHAPTER VL
CoNie Sections.
8xcT. 1. Definitions 162
2. Properties of the Ellipse 165
Problems relating to the Ellipse 167
3. Properties of the Hyperiwla 168
Problems relatine to Hyperbolas 170
4. Properties of the Parabola 171
Problems relating to the Parabola 173
CHAPTER VIL
PROFEBTISS OF CUETES.
Sect. 1. Definitions 175
2. The Conchoid 176
a The Cissoid 177
4. The Cycloid, and Epicycloid 178
6. The Quadratriz 179
& The Catenaiy 180
tenaiy .
I of Relat
Tables of RelatioDs of Catenarian Curves .... 185
PAET n.— MIXED MATHEMATICS.
CHAPTER I.
Mechanics in Oxneeal 187
CHAPTER II.
Statu».
Sect. 1. Statical Eouilibrium 180
2. Center of Gravi^ 102
3l General application of the Prmciples of Statics to the Equili
brium 01 Structures 196
Equilibrium of Piers or Abutments 196
Pressure of earth against Walls 198
Thickness of Walls 201
Bquilibrinm of Polygons 202
Stability of Arches .' . s . .205
Bqpdilicium of Suspension Bridges 207
h
XVUl CONTENTS.
CHAPTER III.
Dynamics.
PAGE
Sect. 1. Oenenl Definitions 211
2. On the General Laws of Uniform and Variable Motion 212
Motion uniformly Accelerated 214
Motion of Bodies under the Action of Gravity . .214
Motion over a fixed Pulley 216
Motion on Inclined Planes 217
3. Motions about a fixed Center, or Axis 220
Centers of Oscillation and Percussion 220
Simple and Compound Pendulums 221
Center of Gyration, and the Principles of Rotation . 229
Central Foroes 232
Inquiries connected with Rotation and Central Forces . 234
4. Percussion or Collision of Bodies in Motion .... 236
5. On the Mechanical Powers 239
Levers 240
Wheel and Axle 242
Pulley 243
Inclined Plane 244
Wedge and Screw 243
CHAPTER IV.
Hydrostatics.
Sect. 1. General Definitions 248
2. Pressure and Equilibrium of Nonelastic Fluids .... 249
3. Floating Bodies 254
4. Specific Gravities 256
5. On Capillary Attraction 258
CHAPTER V.
Hydrodynamics.
Sect. 1. Motion and Effluence of Liquids 260
2. Motion of Water in Conduit Pipes and Open Canals, over
Weirs, «6C 262
Velocities of Rivers 265
3. Contrivances to Measure the Velocity of Running Waters . . 273
CHAPTER VI.
Pneumatics.
Sect. 1. Weight and Equilibrium of .Air and Elastic Fluids . . . 278
2. Machines fur Raising Water by the Pressure of the Atmosph ere 281
3. Force of the Wind 292
CONTENtS. Xix
CHAPTER VIL
BIechanical Agents.
PAGE
Sect. 1. Water as a Mechanical Af^nt 294
2. Air as a Mechanical Agent 297
Coulomb's Experiments 801
dw Mechanical Agents depending upon Heat The Steam Engine . 802
Table of Pressure and Temperature of Steam . . .811
General Description of the Mode of Action of the Steam Engine 812
Theory of the Steam Engine 816
Description of the various kinds of Engines, and the Formulffi
for calculating their Power 822
Practical application of the foregoing Forroulie . 848
4. Animal Strength as a Mechanical Agent 848
CHAPTER VIIL
Strength of Materials.
SaT. 1. Results of Experiments, and Principles upon which they should
be practically applied 870
2. Strength of Materials to Resist Tensile and Crushing Strains 874
Strength of Columns 879
3. Elasticity and Elongation of Bodies subjected to a Crushing or
Tensile Strain 881
4. On the Strength of Materials subjected to a Transverse Strain 885
Longitudinal form of Beam of uniform Strength . 886
Transverse Strength of other Materials than Cast Iron . 887
The Strength of Beams according to the manner in which the
Load is distributed 888
a. Elasticity of Bodies subjected to a Transverse Strain . . 888
6. Strength of Materials to resist Torsion 892
APPENDIX.
L Table of Logarithmic differences 1
n. Table of Logarithms of Numbers, from 1 to 100 .... 1
HL Table of Logarithms of Numbers, from 100 to 10,000 ... 1
IV. Table of Logarithmic Sines, Tangents, Secants, &c. .19
V. Table of Usoiil Factors, extending to several places of Decimals . 64
VL Table of various Useftil Numbers, with their Logarithms ... 66
VIL A Table of the Diameters, Areas, and Circumferences of Circles
and also the Sides of Equal Squares 66
VIII. Table of the Relations of the Arc, Abscissa, Ordinate and Subnormal,
in the Catenary 78
IX. Tables of the Lengths and Vibrations of Pendulums ... 76
X. Table of Specific Gravities 80
XL Table of Weight of Materials frequently employed in Construction . 82
XIL Principles of Chronometers 82
XIII. Select Mechanical Expedients 89
XIV. Obaervarions on the Effect of Old London Bridge on the Tides, &c. 92
XV. Professor Parish on Isometrical Perspective Wi
LIST OF PLATES.
Platb L'
IL
IIL V General Diafframs.
IV. ^
V.
VI. Detiili of a Breast Waterwheel.
VI L Fenton, Murray, and Wood*a Steam Engine.
VII L Higbprenure Expansive Engine, by Middleton.
IX. Details of Steam Engines.
X. Longitudinal Section of a Locomotive Engine.
XL Two Transferee Sections of the Same.
XIL Details of various Engines.
XIIL Isometrical Perspective.
COMMONPLACE BOOK,
ETC., ETC.
PART I.
PURE MATHEMATICS.
CHAP. I.
ARITHMETIC.
Sect. I. Definitions and Notation.
Arithmetic is the science of numbers.
We give the name of number to any assemblage of units^ or of any
parts of an assumed unit; a unit being the quantity which, among
all those of the same kind, forms a whole which may be regarded as
the hose or element. ' Thus, when we speak of one kouse^ one guinea^ ^
we speak of units^ of which the first is the thing called a house, the
fiecond that called a guinea. But when we w,y four houses^ ten
gmnea^^ three quarterM ofaguinea^ we speak of numbers^ of which
the first is the unit house repeated four times; the second is the unit
QHinea repeated ten times; the third is the fourth part of the unit
guinea ^peated three times.
In cv«y particular classification of numbers, the unit is a measure
taken aroitrarily, or established by usage and convention.
Numbers formed by the repetition of an unbroken unit are called
vhole numbers^ or integers^ as seven miles, thirty shillings: those
which are formed by the assemblage of any parts of a unit are called
fractional numbers^ or %\mi^\yf radians; as two thirds of a yard, three
eighths of a mile.
When the unit is restricted to a certain thing in particular, as one
man^ one horse^ one pounds the collection of many of those units is
called a concrete number, aa ten men^ twenty horses^ fifty pounds.
But if the unit does not denote any particular thing, and is expressed
^mply by one^ numbers which are constituted of such units are deno
minated dieerete or ahstracty as five^ ten, thirty. Hence, it is eyident
that abstract nmnbers can only be compared with their unit, as con
B
ARITHMETIC : NOTATION.
[part I.
Crete numbers are compared with, or measured by, theirs; but that it
is not possible to compare an abstract with a concrete number, or a
concrete number of one kind with a concrete number of another ; for
there can exist no measurable relations but between quantities of the
same kind.
The series of numbers is indefinite; but only the first nine of them
are expressed by different characters, called figures; which are as
follows : —
Names, one, two, three, four, ^sc^ six, seven, eight, nine.
Figures. 1, 2, 3, 4, 5, C, 7, 8, 9.
These are called sig?iificant figures, in contradistinction to another
character employed, namely 0, called the cypher or zero; which has
no particular value of itself, but by its positmi is made to change the
value of any significant figures with which it is connected.
In the system of numeration now generally adopted, and which is
borrowed from the Indians*, an infinitude of words and characters is
* As the Roman notation is not unfrequently met with, especially in dates,
we subjoin the following brief account of it: — The Romans employed only
seven numerals, being the following capital letters of their alphabet ;— viz. I,
for one ; V, for five; X, for ten; L, for fifty ; C, for an hundred ; D, for five
hundred; M, for k thousand ; and for expressing any intermediate or greater
numbers they employed various repetitions and combinations, the principles of
which are shown in the following examples : —
As often as any character is repeated, so
many times is its value repeated.
A less character before a greater diminishes
\ its value.
( A less character after a greater increases
( its value.
I
ir
III
4 = nil or IV
5
6 <
7 •
8
9
10
40
50
m
100
500
1000
2000
5000
0000
10,000
50,000
00,000
100,000
1,000,000
2,000,000
&c.
V
VI
VII
VIII
IX
X
XL
L
LX
C
Dor lo
M or Clo
MM
V or loD
yi
X or CCI33
W I J33
LX
CorCCCl033
M or CCCCI3303
MM
&c.
5 For every f) annexed, this becomes ten
( times as many.
\ For every C and ^), placed one at each
( end, it becomes ten times as much.
S A bar over any num1)er increases its value
\ 1000fold.
CHIP. I.] arithmetic: notation. 3
avoided, by a simple yet most ingenious expedient, which is this : —
(urjifyHre placed to t/te left of another assumes ten times the wUue
(to \i rtoM have if it occupied the place of the latter.
Thus, to express the number that is the sum of 9 and 1, or ten
units, (called ten^) we place a 1 to the left of a 0, thus 10. So again
the sum of 10 and 1, or eleven^ is represented by 11 ; the sum of 11
ud 1, or of 10 and 2, (called twelWy) is represented by 12 ; and so
on for tiwieen, fourteen^ fifteen^ &c., denoted respectively by 13, 14,
15, &c, the figure 1 being all along equivalent to ten^ because it
oecapies the second rank.
la like manner, twenty^ twenty^one^ twentytwo^ &c., are repre
sented by 20, 21, 22, because the 2 in the second rank is equivsJent
to twice ten, or tteenty. And thus we may proceed with respect to
the numbers that fall between twenty and three tens or ^irty (30),
four tens or forty (40), five tens or Jifily (50), six tens or sixty (60),
RTen tens or seventy (70), eight tens or eiyhty (80), nine tens or
iit)ie^(90). After 9 are added to the 90 (ninety), numbers can no
longer be expressed by two figures, but require a third rank on the
left hand of tbe second.
Tbe figure that occupies the third rank, or of hundredths^ is ex
pressed by the word hundred. Thus 369, is read three hundred and
sixtjnine ; 428, Is read four hundred and twentyeight ; 837, eight
kundred and thirtyseven : and so on for all numbers that can be re
presented by three figures.
Bat if the number be so large that more than three figures are
required to express it, then it is customary to divide it into periods of
tkree figures each, reckoning from the right hand towards the \eh, and
to distinguish each by a peculiar name. The second period is called
tbat of thousands^ the third that of millions^ the fourth that oi milliards
or billions*^ the fifth that of trillions^ and so on ; the terms units, tens,
ud hundreds, being successively applied to the first, second, and
third ranks of figures from the right towards the left, in each of these
periods.
Thus, 1,111, is read one thousand one hundred and eleven.
23,456, twentythree thousands, four hundred and fiftysix.
421,835, four hundred and twentyone thousands, eight hundred
•nd thirtyfive.
732,846,915, seven hundred and thirtytwo millions, eight hundred
tnd fortysix thousands, nine hundred and fifteen.
The manner of estimating and expressing numbers which we have
here described is conformable to what is denominated the decimal
notation. But, besides this, there are other kinds invented by philo
sophers, and others indeed in common use : as the duodecimal^ in
' It has been cascmnary in England to give the name of LiHums to millions
of millions, of triUknu to millions of millions of millions, and so on : but the
method here given of dividing numbers into periods of three figures instead of
six, it universal (m the Continent ; and, as it seems more simple and uniform
tian the other, I have adopted it.
B 2
4 arithmetic: notation. [part i.
which every superior name contains (tcelve units of its next inferior
name ; and the sexagesimal^ in which sixty of an inferior name are
equivalent to one of its next superior. The former of these is em
{)loyed in the measurement and computation of artificers' work ; the
atter in the division of the circle, and of an hour in time.
To the head of notation we may also refer the explanation of the
principal symbols or characters employed to express operations or
results in computation. Thus,
The sign + (plus) belongs to addition, and indicates that the
numbers between which it is placed are to be added together. Tlius,
5 + 7 expresses the sum of 5 and 7, or that 5 and 7 are to be added
together.
The sign — (mintis) indicates that the number which is placed after
it is to be subtracted from that which precedes it. So, 9—3 denotes
that 3 is to be taken from 9.
The sign ^^ denotes diffeience^ and is placed between two quan
tities when it is not immediately evident which of them is the greater.
The sign x {into\ for multiplication, indicates the product of two
numbers between which it is placed. Thus 8x5 denotes 8 times 5,
or 40.
The sign f {hif)^ for division, indicates that the number which
precedes it is to be divided by that which follows it ; and the quotient
that results from this operation is often represented by placuig the
first number over the second with a small bar between them. Thus,
15 r 8 denotes that 15 is to be divided by 8, and the quotient is ex
pressed thus y .
The sign =, two equal and parallel lines placed horizontally, is
that of equality. Thus, 2  3 + 4 = 9, means that the sum of 2, 3,
and 4, is equal to 9.
Inequality is represented by two lines so drawn as to form an angle,
and placed between two numbers, so that the angular point turns to
wards the least. Thus, 7^4, and A > B, indicate that 7 1^ greater
than 4, and the quantity represented by A greater than the quantity
represented by B ; and, on the other hand, 3 < 5 and C < D indicate
that 3 is less than 5, and C less than D.
Colons and double colons are placed between quantities to denote
their proportionality. So, 3 : 5 : : 9 : 15, signifies that 3 are to 5
as 9 are to 15, or  zi yj.
The extraction of roots is indicated by the sign v/, with a figure
occasionally placed over it to express the degree of the root ; or by a
fraction (having unity for its numerator, and the figure expressing the
degree of the root for its denominator) placed above and to the right
of the quantity to have its root extracted ; thus, v/ 4 or 4*, signifies
the square root of 4 ; V27, or 27*, the cube root of 27; V 16 or 16*,
the fourth or biquadrate root of 1 6.
The raising of powers is expressed by a whole number similarly
placed, the figure denoting the, power to which the quantity is to be
raised ; thus 6'. signifies the square of 6 ; 8', the cube of 8 ,* and 3\
CHAP. I.] ADDITION OF WHOLE NUMBERS. 5
the fourth power of 3. The figures thus used to indicate the power,
whether whole or/ractional numbers, are termed indices or exponents.
When both operations are to be successively performed upon a
quantity, that is, when some root is to be extracted, and then that root
to be raised to some different power, the operation is very simply ex
pressed by a fraction placed as before, the denominator of which indi 
cates the root to be extracted, and the numerator the power to which
that root is to be afterwards raised. Thus 27*, denotes that the cube
root of 27 is to be extracted, and that the root so obtained is to be
squared, or raised to the second power. It is immaterial which
operation is first performed, for the result would be the same whether
we first extracted the cube root, as above, and then squared it^ or
whether we squared the number first, and then extracted the cube root
of the power so obtained.
Although the above signs * are principally employed in Algebra
and the higher branches of Mathematics, they are given here, as their
use in Arithmetic frequently affords brevity without a sacrifice of
perspicuity.
Sect. II. Addition of Whcle Numbers,
Addition is the rule by which two or more numbers are collected
into one aggregate or sum.
Suppose it were required to find the sum of the numbcre 3731,
349, 12487, and 54. It is evident that if we computed separately .
the sums of the units, of the tens, of the hundreds, of the thousands,
&c^ their combined results would still amount to the same. We
should thus have 15 thousands + 14 hundreds + 20 tens + 21 units,
or 15000 + 1400 + 200 + 21 ; operating again upon these, in
like manner, rank by rank, we should have 1 thousands + 6 thou
sands + 6 hundreds + 2 tens + 1, or 16621, which is the sum
required.
But the calculation is more commodiously effected in the following
manner : —
Rtde. — Place the given numbers under each other, so that units
stand under units, tens under tens, hundreds under hundreds, &c.
Add up all the figures in the column of units, and observe for every
ten in its amount to carry one to the place of tens in the second column,
putting the overplus figure in the first column. Proceed in the same
manner with the second column, then with the third, and so on till
alJ the columns be added up : the figures thus obtained in the
several amounts indicate, according to the rules of notation, the sum
required.
Nate. — Whether the addition be conducted upwards or downwards,
the result will be the same ; but the operation is most frequently con
ducted by adding upwards.
* There are other signs employed in the processes of Algebra, an explanation
of which will Ue found at page 52, st icq.
6 SUBTBACTION OF WHOLE NUMBBBS. [PABT I.
Example. — Taking the same numbers as before, and 3731
disposing them as the rule directs, we have 4 + 7 + 9 + 349
1 =z 21, of which we put down the 1 in the place of uniU^ 12487
and carry the 2 to the tens : then 2h5 + 8 + 4f3= 54
22, of which we put down the left hand 2 in the place of
tens^ and carry the other to the hundreds : then 2 + 4 + 1662 1
3 + 7 = 16, of which the 6 is put in the hundreds^ and =*^
the 1 carried to the thousands. This progress continued will give the
same sum as before.
Other Examples,
57
762
5389
97615
III.
6475
9830
2764
5937
77756
3388
9763
90257
10376786
789632
1589
73
103823
25006
181164
11168080
Sect.
Subtraction Oj
f mole Numbers.
Subtraction is the rule by which one number is taken from
another, so as to show the difference, or excess.
The number to be subtracted or taken away is called the subtrahend ;
the number from which it is to be taken, the minuend; and the quan
tity resulting, the remainder.
Rule. — Write down the minuend, and beneath it the subtrahend,
units under units, tens under tens, and so on. Then beginning at the
place of units, take each figure in the subtrahend from its correspond
ing figure in the minuend, and write the difference under those figures
in the same rank or place.
But if the figure in the subtrahend be greater than its corresponding
figure in the minuend, add ten to the latter, and then take the figure
in the subtrahend from the sum, putting down tlie remainder, as be
fore ; and in this case add 1 to the next figure to the le/t in the sub
trahend, to compensate for the ten borrowed in the preceding place.
Thus proceed till all the figures are subtracted.
Example: . . Minuend 26565874
Subtrahend 9853642
Remainder 16712232
Here the five figures on the right of the subtrahend are each less
than the corresponding figures in the minuend, and may therefore be
taken from them, one by one. But the sixth figure, viz. 8, cannot be
taken from the 5 above it. Yet, as a unit in the seventh place is
equivalent to 10 in the sixfhy this unit borrowed (for such is the
CHAP, I.] MULTIPLICATION OP WHOLB NU1IBBB8. 7
technical word here employed) makes the 5 become 15. Then 8
taken from 15 leaves 7, which is put down ; and 1 is added to the 9
in the seventh place of the subtrahend, to compensate or balance the
1 which was borrowed from the seventh pkwjc in the minuend. Re
course must be had to a like process whenever a figure in the sub
trahend exceeds the corresponding one in the minuend.
Other Examples,
From 8217 From 44444 Take 21498 Take 45624
Take 3456 Take 3456 From 76262 From 80200
Remains 4761 Remains 40988 Remains 54764 Remains 34576
iVbte. — Although it is customary to place the minuend above the
subtrahend, this is not absolutely necessary. Indeed, it is often con
venient in computation to find the difference between a number and
a greater that naturally stands beneath it : it is, therefore, expedient
to practise the operation in both ways, so that it may, liowever it
occurs, be performed without hesitation.
Sbct. IV. MuUiplication of Whole Numbers.
Multiplication of whole numbers is a rule by which we find
what a given number will amount to when it is repeated as many
times as are represented by another number *.
The number to be multiphed, or repeated, is called the mtUiipli
candj and may be either an abstract or a concrete number.
The number to be multiplied by is called the multiplier^ and mtist
be an abstract number, because it simply denotes the number of times
the multiplicand is to be repeated.
Both multiplicand and multiplier are called yac^or«.
The number that results from the multiplication is called the pro
duct.
Before any operation can be performed in multiplication, the
learner must commit to memory the following table of products, from
2 times 2, to 12 times 12.
* This definition, though not the most scientific that might be given, is placed
Ymbt^ because others depend, implicitly if not explicitly, on proportion, and
Iharafore cannot logically be introduced thus early in the course.
MULTIPLICATION OF WHOLE NUMBERS.
[part I.
times
2
3
4
5
6
7
8
9
10
11
12
24
2
4
6
8
10
12
14
16
18
20
22
3
4
6
9
12
15
18
21
24
27
30
33
36
8
12
16
20 1 24
28
32
36
40
44
48
5
10
15
20
25
30
35
40
45
50
55
60
6
12
18
24
30
36
42
48
54
60
66
72
7
14
21
28
35
42
49
56
63
70
77
88
84
96
8
16
24
32
40
48
56
64
72
80
9
18
27
36
40
45
54
63
72
81
90
99
108
10
20
30
50
60
70
80
90
100
110
120
11
22
33
44
55
66
77
88
99
110
121
132
144
12
24
3G
48
60
72
84
96
108
120
132
It is very advantageous, in practice, to have this table carried on,
at least intellectually, to 20 times 20. All the products to this extent
are easily remembered.
The learner will perceive that in this table 7 times 5 is equal to
5 times 7, or 7 X 5 =: 35 = 5 X 7. In like manner that
8 X 3 = 24 = 3 X 8, 4 X 11 = 44 z= 11 X 4, and so of other
products. This is often made a subject of formal proof, as well as that
3x5x8 — 3x8x5 = 5x3x8 = 5x8x3, &c. But
to attempt the demonstration of things so nearly axiomatical as these
is quite unnecessary.
Previously to exhibiting the rules for performing multiplication, let
us take a simple example, and multiply 4827 by 8. Here
placing the numbers as in the margin, and multiplying in
their order 7 imits by 8, 2 tens by 8, 8 hundreds by 8, 4
thousands by 8, the several products are 56 units, 16 tens,
6.4 hundreds, 32 thousands : these placed in their several
ranks, according to the rules of notation, and then added up,
give for the sum of the whole, or for the product of 4827
multiplied by 8, the number 38616.
4827
8
56
16
64
32
38616
CHAP. I.] MULTIPLICATION OP WHOLE NUMBERS. 9
Or the same example may be worked thus : —
8 X 7 = 56 \
Sx 800= eJoo/ ^^^"^^^«^I^^^°^^3^i*^^
8 X 4000 = 32000 i f™® '" ^"^^^ ^ ^'''
I fore.
38G16 /'
Cask I. — To multipltf a number^ consisting of several figures^ by a
Mumher not exceeding 1 2.
Ruk. — Multiply each figure of the multiplicand by the multiplier,
beginning at tlie units ; write under each figure the units of the pro
duct, and carry on the tens to be added as units to the prodqct fol
lowing.
Examples,
Mnltiplv
4827
218043
440052
8765400
Bv '
8
9
11
12
Products 38616 1962387 4840572 105184800
Case II. — To perform multiplication token each /actor exceeds 1 2.
Rule. — Place the factors under each other (usually the smallest at
bottom), and so that units stand under units, tens under tens, and so
on. Multiply the multiplicand by the figure which stands in the unit's
place of the multiplier, and dispose the product so that its unit's place
^hftll stand under the unit of the multiplicand ; then multiply succes
sixely by the figure in the place of tens, hundreds, &c., of the multi
plier, and place the first figure of each product under that figure of the
moltiplicr which gave the said product. The sum of these products
will be the product required.
Example,
Multiply 8214356 by 132.
Multiplicand 8214356
Multiplier 132
8214356 X 2 = 16428712
8214356 X 3 tens = 24643068
8214356 X 1 hundred = 8214356
8214356 X 132 = 1084294992
10 DIVISION OF WHOLB NUMBBB8. [PART I.
Other Example$.
Multiply 821436 Multiply 8210075
by 672576 by 420306
4928616 49260450
5750052 24630225
4107180 16420150
1642872 32840300
5750052
4928616 Product 3450743782950
Product 552478139136
Note, — Multiplication may frequently be shortened by separating
the multiplier into its component parts or factors, and multiplying by
tbem in succession. Thus, since 132 times any number are equal to
12 times 11 times that number, the first example may be performed
in this manner :
Multiply 8214356
by 11
Here one line of multi
And this product 90357916 ,» plication, and one of
by 12 i addition, are saved.
Product as before 1084294992 /
So, again, the multiplier of the second example, viz. 672576,
divides into three numbers, 600000, 72000, and 576; where, omitting
the cypher, we have 72 =: 12 X 6, and 576 = 8 X 72. Hence
the operation may be performed thus: —
Multiplicand 821436
Multiply by 6 in the 6th place.
4928616
Previous product X 12 ... 59143392 for 72 thousands.
Second product X 8 473147136 for 576 units.
Same product as before 552478139136 : three lines saved.
Other modes of contraction will appear as we proceed.
Sbct. V. Division of Whole Numbers.
Division is a rule by which we determine how often one number
is contained in another. Or, it is a rule by which, when we know a
product and one of the factors which produced it, we can find the
other.
The number to be divided in called the dividend ; that by which it
CHAP. I.] ]>I VISION OP WHOLB NUMBBRB. 11
is diTided, the divisor; and that which results from the divisioD, the
([ua^tnt. When division and multiplication are regarded as reciprocal
operations, the dividend is equivalent to the product^ the divisor is
equiTalent to the tntdtiplier^ and the quotient is equivalent to the mtd
tipUeand.
Rule. — Draw a curved line both on the right and left of the dividend,
and place the divisor on the left ; then find the number of times the
dirisor is contained in as many of the lefthand figures of the dividend
as are jost necessary, and place that number on the right. Multiply
the divisor by that number, and place the product under the above
mentioned figures of the dividend. Subtract the said product from
that part of the dividend under which it stands, and bring down the
Bext figure of the dividend to the right of the remainder. Divide the
remainder thus increased, as before ; and if at any time it be found
less than the divisor, put a cypher in the quotient, bring down the
next figure of the dividend, and continue the process till the whole is
finished : the figures thus arranged will be the quotient required.
Examples,
Divide 743256 by 324.
Dividend.
Divisor 324)743256(2294 Quotient.
648 Divisor 324
Quotient 2294
648
3045
2916
Pro
131)135076(1031^^^: '
131
1296
2916
648
648
1296
1296
Remain
71)29754(419^
284
►of 743256
In these two ex%
amples the num
bers which re
main are placed
ot^ their respect
ive divisors, and
"attached to the
quotients ; the
meaning of which
will be expbiined
when we treat of
135
71
644
639
5 Remain.
407
393
146
131
15 Remain.
Note.^Wheu the divisor does not exceed 12, the operation may
readily be perfoimed in a single line; as will appear very evident if
12
DIVISION OF WHOLE NUMBERS.
[PABT I.
tbc following example be compared with the two methods of working
the first example in multiplication.
Divide 38616 by 8.
8)38616(4827 Dividend 38616
32 Divisor 8
66
64
Quotient 4827
21
16
56
56
Here 38 contains 8 four times, leaving a remainder
of 6; these carried as 6 tens to the next 6, make
66, which contains 8 eight times, leaving 2, which
carried as 2 tens to the next figure 1, make 21 : and
so of the rest.
In division, also, upon the same
principle as in multiplication, the la
bour may often be abridged by taking
component parts of the divisor. Thus,
in the first example, the divisor is equal
to 4 times 81, or 4 times 9 times 9.
Hence the dividend may be divided by
4, 9, and 9, successively, as in the
margin, and the result will be the same
as before.
Divide 743256
by 4
this quotient 185814
by 9
and this 20646
by 9
Quotient 2294
Since 25 is a fourth part of 100, and 125 the 8th part of 1000,
it will be easy to multiply or to divide by either of these numbers iu
a single line — thus.
To multiply 4827 by 25, put two
cyphers on the right, which is
equivalent to multiplying by 100;
and divide by 4.
4)482700
120675 Answer.
To divide 582100 by 25, strike
ofi^ two figures on the right hand,
which is equivalent to dividing by
1 00 ; then multiply by 4.
5821 100
4
To multiply 6218 by 125, put 3
cyphers, which is equivalent to
multiplying by 1000; then di
vide by 8.
8)6218000
777250 Answer.
To divide 4567000 by 125, strike
off three figures on the right hand,
which is equivalent to dividing by
1000; then multiply by 8.
45671000
8
23284 Answer.
36536 Answer.
2758
2758
3099
3099
4C9
1029
469
1029
7355
CHAP. I.] VULGAR FRACTIONS. 13
PROOF OF THE FIRST FOUR RULES OF ARITHMETIC.
Simple as these four rules are, it is not unusual to commit errors
in working them : it is, therefore, useful to possess modes of proof.
1. Now, addition may be proved by
adding downwards, as well as upwards,
and observing whether the two sums
agree ; or, by dividing the numbers, to be
added into two portions, finding the sum
of each, and then the sum of those two
separate amounts. Thus, in the margin, ^ss^ 5857
the sum of the four numbers is 7355 ;
the sum of the two upper ones 5857, 1498
of the two lower ones 1498, and their
sum is 7355, the same as before. 7355
2. The proof of subtraction is effected by adding the remainder to
the subtrahend ; if their sum agrees with the minuend the work is
right, otherwise not.
3. Multiplication and division reciprocally prove each other.
There is also another proof for multiplication, known technically
by the phrase casting out the nines. Add together the numbers from
left to right in the multiplicand, dropping 9 whenever the sum exceeds
9, and carry on the rem£under, dropping the nines as often as the
amount is beyond them ; and note the last remainder. Do the same
i^ith the multiplier and with the product ; then multiply the first two
remainders and cast the nines out of their product; if the remainder
is equal to the last remainder, this is regarded as a test that the work
is right. Thus, taking the second example in multiplication, the
figures in the multiplicand amount to 6 above two nines, those in the
multiplier to 6 above three nines, those in the product to above six
nines ; the product 6 x G of the two first excesses is 36, or above
four nines : the coincidence of the two O's is the proof. It is plain,
however, that the proof will be precisely the same so long as the
figures in the product be the same, whatever be their order : the
proof, therefore, though ingenious, is defective •.
A similar proof applies to division.
Sect. VI. Vulgar Fractions.
The fractions of which we have already spoken in Sect. L, are
usually denominated Common or Vulgar Fractions, to distinguish
* The correctness of this proof, with the exception above specified, may be
shown algebraicaUy, thus : — put M and JV — the number of nines in the mul
tiplicand and multiplier respectively, m and n their excesses ; then, 9 M •\ m =^
the multiplicand, and 9 Jv f n » the multiplier, and the product of those
factors will l>e — 81 M N + 9 Af n + 9 iV m = m n ; but the three first
terms are each a precise number of nines ; because one of the factors in each
is so ; these, therefore, being neglected, there remains m n to be divided by nine ;
but m n is the product of the two former excesses : therefore the truth of the
method is evident. Q> E. D.
14 VULGAR FRACTIONS. [PART I.
them from another kind, hereafter to he mentioned, called DecinuU
Fractions,
A fraction is an expression for the value of any part of an integer,
or whole numher, such numher heing considered as unity. Thus, if
a pound sterling he the unit, then a shilling will he the twentieth
part of that unit, and /our pence will he four twelfths of that twen
tieth part. These represented according to the usual notation of
vulgar fractions, will he ^^ and ^^ of ^^ respectively.
The lower numher of a fraction thus represented (denoting the
numher of parts into which the integer is supposed to he divided) is
called the denominator ; and the upper figure (which indicates the
numher of those parts expressed by the fraction) the numerator.
Thus, in the fractions ^, j^, 7 and 15 are denominators^ 5 and 8
numerators.
Vulgar fractions are divided into proper, improper, mixed, simple,
compound, and complex.
Proper fractions have their numerators less than their denomi
nators, as ^, ^, &c.
Improper fractions have their numerators equal to, or greater than,
their denominators, as ^, i^, &c.
Mixed fractions, or numbers, are those compounded of whole
numbers and fractions, as 7^9 12^, &c.
Simple fractions are expressions for parts of whole numbers, as
%, ^ &c.
Compound fractions are expressions for the parts of given fractions,
as I of I, ^ of 3?^, &c.
Cimijiex fractions have either one or both terms fractional
52 12 6^
numbers, as ^^ j^, ^^ &c.
The value of a fraction is not altered by multiplying or dividing
both its numerator and denominator by the same number; thus
i? \'> W^ Jli' *^® ^ equal, although successively multiplied by 2, 6,
and 12.
Any number which will divide two or more numbers without
remainder, is called their common measure.
REDUCTION OF VULGAR FRACTIONS.
This consists principally in changing them into a more commodious
form for the operations of addition, subtraction, &c.
Case I. — To reduce fractions to their lowest terms,
Rtde, — Divide the numerator and denominator of a fraction by any
number that will divide them both, without a remainder ; the quotient
again, if possible, by any other number : and so on, till 1 is the
greatest divisor.
Thus, m^ =z If* = ^\\ = i\ = I, where 5, 3, 7, 7, re
spectively, are the divisors.
^"■5 IHi = h ^7 dividing at once by 735.
CHAP. I.] VULGAR FRACTIONS. 15
Nnfte, — This number 735 is called the greatest common measure of
the terms of the fraction : it is found thus — Divide the greater of the
two numbers by the less; the last divisor by the last remainder, and
so, on till nothing remains : the last divisor is the greatest common
measure required*.
Case II. — To redu4X an improper f ration to its equivalent whole or
mixed number.
Rule, — Divide the numerator by the denominator, and the quotient
will be the answer: as is evident from the nature of division.
EtX, — Let ^^ and ^4^ be reduced to their equivalent whole or
mixed numbers.
43)957(22^} Answer. 274)5480(20 Answer.
86 548
97
86 =
11
Cask III. — To reduce a mixed number to its equivalent improper
Jraction ; or a whole number to an equivalent fraction having any
assigned denominator.
Rule, — This is, evidently, the reverse of Case II.; therefore multiply
the whole number by the denominator of the fraction, and add the
numerator to obtain the numerator of the fraction required.
Ex, — Reduce 22:^ to an improper fraction, and 20 to a fraction
whose denominator shall be 274.
(22 X 43) + 11 = 957 new numerator, and \^ the 1st
fraction.
20 X 274 z= 5480 new numerator, and ^^ the 2nd fraction.
* The foUowing theorems are useful for abbreviating Vulgar Fractions :—
Theorems.
1. If the last digit of any number be divisible by 2, the whole number is
divisible by 2. If the two last dig:its be divisible by 4, the whole number is
divisible by 4. If the three last digiu be divisible by 8, the whole number is
divisible by 8. And, generally, if the last n digits of any number be divisible by
2*, the whole number is divisible by 2*^.
2. If a number terminate with 5, it is divisible by 5 ; and if it terminate in
0, it is divisible by either 10 or o.
3. If the sum of the digits constituting any number be divisible by 3 or 9, the
whole is divisible by 3 or 9 ; and if also the last digit be even, the whole number
is divisible by 18.
4. If the sum of the digits constituting any number be divisible by 6, and the
right.hand digit by 2, the whole is divisible by 6 : for by the data it is divisible
both by 2 and 3.
5. If the sum of the 1st, 3rd, 5th, &c , digits constituting any number be equal
to that of the 2nd, 4th, 6th, Ac., that number is divisible by 11 : for if a, 6, c,
(f , e, m, n, be the digits, constituting any number, its digits, when multiplied
by 11, will become
(8) (7) (6) (5) (4) (3) (2) (1)
«, a+4, A+c, c+rf, rfh^, cfm, m^n^ n ;
where the odd temns are » to the even.
id VULGAR FRACTIONS. [pART I.
Case IV. — To reduce a compound frartion to an equivalent simple
one.
Rule, — Multiply all the numerators together for the numerator, and
all the denominators together for the denominator, of the simple
fraction required.
If part of the compound fraction be a mixed or a whole number,
reduce the former to an improper fraction, and make the latter a
fraction bv placing 1 under the numerator.
When like factors are found in the numerators and denominators,
cancel them both.
Ex,— Reduce ? of \ of ^ of ^ of ,8^ to a simple fraction.
2x3xr> x7x8 2x5x8 1x6x8 1x5x4 20
3X4X7X9 Xll""! X 9 X ll""2 X 9 X ll""! X 9 X ll^OQ
Here the 3 nnd 7 common to numerator and denominator are first
cancelled ; then the fraction is divided by 2 ; and then by 2 again.
/t^x.— Reduce throe farthings to the fraction of a pound sterling.
A farthing is the fourth of a penny, a penny the twelfth of a shilling,
nnd a shilling the twentieth of a pound.
Therefore ^ of f^y of ^ zz ^J^ = ^^^y the answer.
2^
Ex, — Simplify the complex fraction ...
H
Here, reducing the mixed numbers to improper fractions, we have
8
— : multiplying by 3, to get quit of the denominator of the upper
V
8
fraction, we have p: multiplying by 5, to get quit of the denominator
of the lower fraction, we have ^J: dividing both terms of this fraction
by 8, there results ^ for the simple fraction required.
Case V. — To reduce fraction% of different detioininators to equi
valent fractions having a common detiominaior.
Rule. — Multiply each numerator into all the denominators except
its own, for new numerators ; and all the denominators together for a
common denominator.
Ex, — Reduce , ^, and §, to equivalent fractions having a common
denominator.
2 X 7 X 9 = 126 \
6 X 3 X 9 = 162 [ the numerators.
5 X 3 X 7 = 105 )
3 X 7 X 9 = 189, the common denominator.
Hence the fractions are [^, ^, ^g^, or *«, * J, ^^, when divided
by 3.
Hence, also, it appears that f exceed §, and that 4 exceed ^,
Ex, — Reduce  of a penny, and J of a shilling, each to the fraction
of a pound ; and then reduce the two to fractions having a common
denominator.
* of a penny = ^ of ^^ of 4^ = p*,^^ = ^^^ of a pound.
% of a shilling = ^ of ^^ = ^% = ^ = ^ of a pound.
Hence ^ of a shilling are 1 times as much as  of a penny.
CBiP. I.] VULOAB FRACTIONS. 17
Mute.— Other methods of reduction will occur to the student after
tolenble practice, and still more after the principles of algebra are
ttqaired.
ADDITION AND SUBTRACTION OP FRACTIONS.
RuU,—lf the fractions have a common denominator, add or sub
tract the numerators, and place the sum or difference as a new numc
ntor oTer the common denominator.
If the fractions have not a common denominator, they must be re
duced to that state before the operation is performed.
In addition of mixed numbers, it is usually best to take the sum of
tbemtegere, and that of the fractions, separately ; and then their sum,
for the resolt required.
Examples.
1. Find the sum of f , ^, and .
. l;f ^ t* "TP + H + fi = V»' = 2H
z. Take  of a shilhng from y^ of a pound sterhng.
I of a shilling =  of ^>„ = ^^ of a pound = ^q.
Also rV of a pound = /^j. Hence ^%  ^^^ = ^^Vff
= ^»j = 11 pence.
3. Find the difference between 12 J and 8 J.
»2 H = V  V = W  Vo" = SV = 4/»
MULTIPLICATION AND DIVISION OF FRACTIONS.
R*ik 1. To multiply a fraction by a whole number, mtdtiplif the
KMPKrttfor by that number, and retain the denominator : — Or, divide
^dtnomnator by the same number, (if a multiple of it,) and retain
tbe numerator.
2. To divide a fraction by a whole number, multiply the denomi
■•tor by that number, and retain the numerator : — Or, divide the
MJWrotor by the same number, (if a multiple of it,) and retain the
Nominator.
3. To multiply two or more fractions is the same as to take a
friction of a fraction ; and is, therefore, effected by taking the pro
duct of the numerators for a new numerator, and of the denomina
tors for a new denominator. (The product is evidently smaller tlian
eitber factor when each is less than unity.)
*. To divide one fraction by another, invert the divisor, and pro
«wd as in moltiplication. (The quotient is always greater than the
dividend when the divisor is less than unity.)
Examples.
1. Multiply { by 2, and divide ^ hy 5.
c
18
DECIMALS.
[PABT I.
Multiply 25 by f , and divide f by ji
1i, Ans,
3 =8 ^2;andf ^=^
lV =
Afultiply £2 IBs. U. bv 3}, and divide £4 15«. by 3^,
£2 13«. 4</. = 2 + IJ + T*/ of A = 21 = f, and
= 8 X J = V == «/ = 9^ = £9 6«. 8</.
£4 158. H 3i = 4 r 3i = V ^ V = V
liJ = £1 8«. 6</.
i^Tofe. — In the multiplication of mixed
numbers, it is often less laborious to
iV
perform the multiplication of each part
separately, and collect their sum, as in
the margin, than to reduce the mixed
numbers to improper fractions, and re
duce their product back again to a mixed
number.
45
45
8
T
45
17
Multiply 45f
ByJTf
7 =315
1 ten 3= 45 .
I = .30
% = 'jn
Product 808^
Sect. VII. Decimal Fractions,
The embarrassment and loss of time occasioned by the com
putation of quantities expressed in vulgar or ordinary fractions, have
inspired the idea of fixing the denominator so as to know what it is
without actually expressing it. Hence originate two dispositions of
numbers, decimal fractions and complex numbers. Of the latter,
such, for example, as when we express lineal measures in yards, in
feet (or thirds of a yard), and inches (or twelfths of a foot), we shall
treat in the following section, and shall here confine ourselves to the
former.
Decimal fractions, or substantively, decimals^ are fractions always
having some power of ten for their denominator ; but for the sake of
brevity only the numerator is expressed, being written as tL^whok
number with a dot placed on its left hand, which dot determines the
value of the denominator, the number of cyphers in the denominator
being always equal to the number of figures to the right of the dot,
or as it is termed decimal point ; if the number of significant figures
in the numerator is not sufficient, cyphers are added to the left hand.
It is evident that the values of decimals decrease in the same tenfold
proportion from the point to^^ards the right hand, as those of integers
mcrease towards the left : — thus
Igai
r fraction,
tV is written
•1
>i
99
aha 99
•01
»
99
TO (TU 99
•001
>i
99
TTJOcJi) 99
•0001
»>
99
iV >9
•7
»
99
tVo »>
•43
»
99
"rioinr »>
•0125
i«
written
•3
*
>i
•6
4
y»
•428571
m
>f
•29504
CHAP. I.] DECIMALS. 19
The vulgar fraction, 7/^ is written 73
42^*5 „ 4285
57,VoV >> 57217
&c. &c.
The Talae of a decimal fraction is not altered by cyphers on the
Jight hand : for '500, or ^W^, is in value the same as ^^, or 5,
that is .
When decimals terminate after a certain number of figures, they are
called/iii^«,as125 = Vt5V(jpi958 = T^V« = m ^ . .
When one or more figures in the decimal become repeated, it is
oaHed a repeating or circulating decimal ; and a dot is placed over the
figure to be repeated, if only one, or if more than one, over the first
and last figures ; thus : —
•333333, &c.
'666666, &c.
•428571428571, &c.
•29504504, &c.
When the circulating portion of the decimal is preceded by other
figures which do not circulate, (as in the last example,) it is called a
mixed circulate.
Rules for the management of this latter kind of decimals are given
by several authors ; but, in general, it is more simple and commodious
to perform the requisite operations by means of the equivalent
vulgar fractions, the method of obtaining which is given in Case III.
RBDUCTION OP DECIMALS.
Reduction of Decimals is a rule by which the known parts of given
integers are converted into equivalent decimals, and vice verad.
Case I. — To reduce a given vulgar fraction to an equivalent
dedmoL
Rule. — Annex as many cyphers to the numerator as may be neces
sury, then divide by the denominator, and point off in the quotient as
many places of decimals as the number of cyphers added to the
numerator; if the quotient does not contain so many figures, the
deficiency must be made up by cyphers placed on the left hand.
JSxamplea.
1. Reduce ^, , i^, ^, to equivalent decimals.
2) 10 4) 300
•5 decimal =  ; *75 decimal = ^ :
C 4)70000 ( 8)6000
16} «4
^ 4)17500 I 8)^75000
•4375 decimal = f^; '09375 decimal = ^
•asHB  c 2
20 DECIMALS. [PABT I.
2. Reduce ^\ and ^i to equivalent decimals.
/ 3)4000000
27
( 9)1*333333
•148148, &c. =
•148 decimal z= ^\
i 7)110000000
63'
I 9) 15714285714285
•1746031746031, &c =
•174603* decimal = i^.
These two are evidently circulating decimals, in the former of
which the figures 148 become indefinitely repeated, in the latter the
figures 174603.
3. Reduce 14«. 6d, to the decimal of a pound.
First, 14*. ed. = JJ + i of ^ig = IS + ViJ = 18
Then J 8 — 1^" = ''^^^' '^® decimal required.
4. Reduce tJ^ to its equivalent decimal.
57) 44000000 (77192, &c. decimal = ^.
399
170
114
56
Note, — The above fraction is = J x ^, of which the two denomi
nators arc both jortm^ numbers^ (that is, divisible by no other number
than unity,) the entire equivalent decimal is a circulate of 18 places,
t. e. one less than the last prime .... 771929824561403508, 7719,
&c. over again ad infinitum*,
• There are many curioui properties of fractions i « 'i 42867, Ac
whose denominators are prime niimliers, one of I ^ •285714 An
which may lie here shown in reference to fractions * "* . ' .» ^'
having the denominator 7 The circulating fi^nires ? ~ •428671, Ac
of the equivalent decimals are precisely the same, ^ "" 671428, Ac
for \y ), &C., and in the same order: the cir & ^ *714286 Ac
culate merely oommeooes at a different place for each Z a.. , .<:* .
numerator. * * 867142, Ac
CHIP. I.] DECIMALS. 21
Gin II. — Antf decimal being given to find its equivalent vulgar
fnttm; or to express its value bg integers of lower denominafions,
i^.— When the equivalent vdgar fraction is required, place
Qoder the decimal as a denominator a unit with as many cyphers as
there are figures in the proposed decimal ; and let the fraction so con
ititnted be reduced to its lowest terms.
Or, if the value of the decimal he required in lower denominations,
Doltiplj the given decimal hy the number of parts in the next less
deDomioation contained in its integer; and point off, from right to
left, aa many figures of the product as there were places in the given
decimal. Multiply the decimal last pointed off by the value of its
integer, in the next inferior order, pointing off the same number of
dednuJs as before : and thus continue the process to the lowest
bteger, or until the decimals cut off become all cyphers ; then will
tkeseTenJ numbers on the left of the separating points, together with
the remaining decimal, if any, express the required value of the given
decimal.
Examples,
1. Find the vulgar fractions equivalent to '25 and '375.
•25 = ^V^ = i ; and '375 = ^Vrf^^ = , Answers.
2. Fbd the value in shillings, &c. of '528125 of a pound.
•528125
20
10562500 >Ans. lOs. Bid
12 \
67500 = 6
3. Fmd the value of *74375 of an acre.
•74375
4
297500 > Ans. 2 roods 39 perches.
40 ^
39000
Cm III. — To reduce a circulating decimal to its equivalent vtdgar
fi^ttum,
BtJs, — Take the figures in the decimal and place them as a whole
Biimber for the numerator, and under them for a denominator as many
S*! aa there are figures in the circulate, and the fraction thus formed
will be equivalent in value to the given decimal.
If the decimal is a mixed drcukte, subtract the finite part (or the
%iii«s which are not repeated) from the whole mixed circulate, (both
22 DECIMALS. [part I.
considered as whole numbers,) for the numerator; and for the deno
minator, take as many 9's as there arc figures in the circulating por
tion of the decimal, with a8 many cyphers to the right as there are
figures in the finite portion of the same.
Examples.
The circulate 3
= i
i
•06
= ^ =
h
» 549
— wJ *^
^
» 7630
= "rm =
7A
The wwerf circulate 6409 = ^^ll^o*
= il*
529504= 5»^JJ^«5' = 5iH
ADDITION AND SUBTRACTION OF DICIMALS.
These operations are performed precisely as in whole numbers, the
figures being so arranged that units stand under units, tens under tens,
&c., or, ^which amounts to the same thing,) so that the decimal points
stand under one another. Thus,
42175 From 24861 78
Add } 328165 Take 1456789
together i 0027
11 Remains 2471*60511
Sum 4655692 Proof 248617300
MULTIPLICATION AND DIVISION OP DECIMALS.
Here, again, the operations are performed as in integers : Then, in
multiplication, let the product contain as many decimal places as
there are in both the multiplier and multiplicand, cyphers being pre
fixed, if necessary, to make that number ; and, in division, point off
as many decimals in the quotient as the number in the dividend
^including the cyphers supplied, if there be any) exceeds that in the
oivisor.
Examples.
Multiply 437 by 39 1, and 2 4542 by 0053.
437^
391
' ^Here 437 x 391
^^}88S^ = 170^*0%.
as in the decimal
i operation.
24542
•0053
437
3933 .
1311
73626
122710
•01300726
170867
Here one cypher
is prefixed to make
the requisite num
ber of decimals in
the product.
CBIP. I.] COMPLEX NUMBBBS. 23
2. Divide 172*8 by 144, and 192 by 5*423.
•144) 1728 ( 1200 quotient. 5423 ) 192000 ( 3540475
144 16269
288 29310
288 27115
00 21950
= 21692
In the first of these examples, the two 25800
cjphen brought down, together with the deci 21692
mil 8, make the number of decimals in the
diridend the same as in the divisor, there 41080
fore the quotient is composed entirely of in 37961
tegers. In the second example, 3, the num
ber of decimal places in the divisor, taken 31190
from 8, the nnmber in the dividend (including 271 15
thoK brought down), leave 5 for the decimtd
places in the qnotient. 4075
Skt. VIII. Complex Fractions used in the Arts and Commerce.
In the arts and in commerce, it is customary to assume a series of
nnits bsTing a constant relation to each other, so that the units of
ODe denomination become fractions of another. One farthing, for
eumple, is ^ of a penny, 1 penny /.t of a shilling, 1 shilling ^*jy of a
poand, or ^y of a guinea. One lineal inch, again, is yV of a foot, 1 foot
i of A yard ; and so on, according to the relations expressed in the
tible« in Chap. V. on Mensuration, Sect. I. The arithmetical opera
tioDs on complex numbers of these kinds are usually effected by
limpler rules than those which apply to vulgar fractions generally ;
of which it will, therefore, be proper here to specify a few.
BBDUCTION.
Here we have two general cases :
Ca81 I. — When the numbers are to be reduced from a higher de
nomination to a lower : —
Bide. — Multiply the number in the higher denomination by as
■uy of the next lower as make an integer, or one, in that higher,
<id set down the product. To this product add the number, if any,
vhich was in this lower denomination before ; and multiply the sum
hy as Buny of the next lower denomination as make an integer in
tht present one. Proceed in the same manner through all the dcnomi
B*^s to the lowest, and the number last found will be the value
24
COMPLEX NUMBERS.
[part I.
of all tlic numbers whicli were in the higher denominations taken
together.
Case IL — fV/ten the numbers are to be reduced from a lower de
nomination to a higher : —
Rule.—DWidc the given number by as many of that denomination
as make one of the next higher, and set down what remains. Divide
the quotient by as many of this as make one of the next higher de
nomination, and set down what remains in like manner as before.
Proceed in the same manner through all the denominations to the
highest; and the Quotient last found, togetlier with the several re
mainders, if any, will be of the same value as the first number proposed.
The method of proof is to work the question back again.
Examples,
1. Reduce <£l4 to shillings, pence, and farthings; and 24316
farthings into pounds, &c.
14
4)24316
20
280 shillings
12
12) 6079 pence
20) 506 7
3360 pence
4
£25 6«. 7d.
13440 farthings
2. Reduce 22 Ac. 3 R. 24 P. into perches; and 52187 perches
into acres.
a. r. p.
22 3 24
40)52187
4
91 roods
4)1304 27
40
Ac. 326 R. 27 P.
3664 perches.
ADDITION.
/2w/e.— Place the quantities to be added so that those of the same
denomination may be all under each other. Then add up the numbers
in the right hand column, and divide their sum by the number of units
of that denomination contained in an unit of the next denomination
to the left, write the remainder (if any). at the foot of the first column,
and carry the quotient on to the addition of the second column^ and
thus proceed until all have been added up.
CHIP. I.] COMPLEX NUMBSBS. 25
Examj^es.
£ «. dL lb. oar. duft. gr, tb, om. dujt.gr,
'368 10 3 / 14 6 12 13 / 10 8 11 17
1257 10 5 i 17 5 3 12 t 42 5 16 12
.jJ 88 U 4i . ,, J 15 9 16 . , J 12 2 14 18
^^^j 33 10 ^"^"^i 2 7 15 20 ^"^"^ \ 51 6 22
f 12 13 5 f 13 2 10 19 f 24 9 17 17
8 8 8^ \ 4 1 5 21 V 29 4 18 22
Sum 769 4 2 Sam Q^ 11 18 5 Sum 171 2 12
SUBTRACTION.
Ruk, — Write the smaller number beneath the greater, taking care
to keep the same denominations under each other. Then begin at the
right band and subtract the lower number of each denomination from
the upper, writing the remainder underneath.
When the lower number of any denomination is greater than the
upper one of the same, add to the latter as many units as are con
tamed in one unit of the next greater denomination, always taking
care, when such has been done, to add one to the next lower number
to the left.
Examples,
£. M. d. £. g, d, lb, OM,dtoLgr.
From 16 12 8f From 21 13 4J From 18 9 10 8
Take 10 11 sl Take 18 9 8 Take 9 10 15 20
Bern. 6 1 2 Rem. 3 3 8 Rem. 8 10 14 12
MULTIPLICATION.
i^ti/e.— Place the multiplier under the lowest denomination of the
multiplicand. Multiply the number in the lowest denomination by
the moltiplier, and find how many integers of the next higher de
nomiDation are contained in the product, and write down what
remains. — Carry the integers, thus found, to the produce of the next
higher denomination, with which proceed as before ; and so on,
thioagh all the denominations to the highest ; and this product,
together with the several remainders, taken as one number, will be
the whole amount required.
If the multiplier exceed 12, multiply successively by its component
puis; as in the following examples :•—
26 OOMPLEX NUMBKBS. [PABT I;.
ExamfleM,
£ 9m d, a. r. p.
1. Mo]tiplj4 17 e\ by 441, and 3 2 14 by 531.
10
£ $, d.
4 17 Q\ 85 3 20
9 X 7 X 7 = 441 9 10
43
17
lOj
7
307
4
7
An*. £2150
U
lOj
358 3 for 100
5
1793 3 for 500
107 2 20 3 times 10
3 2 14 1 top line.
^n«. 1904 3 34
DIVISION.
Bide. — Place tbe divisor and dividend as in simple division. — Begin
at the left hand, or highest denomination of the dividend, which divide
by the divisor, and write down the quotient. — If there be any re
mainder after this division, find how many integers of the next lower
denomination it is equal to, and add them to the number, if any,
which stands in that denomination. — Divide this number, so found,
by the divisor, and write the quotient under its proper denomination.
— Proceed in the same manner through all the denominations to
the lowest, and the whole quotient, thus found, will be the answer
required.
£ B. d.
2. Divide 521 18 6 by 432.
432 = 12 X 12 X 3.
Therefore, by short division :
12)521 18 6
12)43 9 10^
3)3 12 6i + I a farthing.
Qtwiient £} 4 1 + f of a farthing.
CHAP. I.
DUODECIMALS.
27
By loDg division : —
£ 9. d. £ 9. d.
432)521 18 6 (1 4 1 J + f of a farthing.
432
89
20
432)1798(4
1728
70
12
432)846(1
432
414
4
432)1656(3
1296
m^a^i
360
DUODECIMALS.
Fmetions whose denominators are multiples of 12, as 144, 1728,
&e., ire called duodecimals; and the division and suhdi vision of the
iateger sre Hudersiood without being expressed, as in decimals. The
method of operating by this class of fractions is principally in use
>DiODg artificers, in computing the contents of work, of which the
^eDsions are taken in/eet, inches^ and twelfths of an inch.
Rule. — Set down the two dimensions to be multiplied together,
one under the other, so that feet shall stand under feet, inches under
iocbes, &c. Multiply each term in the multiplicand, beginning at the
West, by the feet in the multiplier, and set the result of each
iou&ediately under its corresponding term, observing to carry 1 for
vvery 12, from the inches to the feet. In like manner, multiply all
tbe multiplicand by the inches of the multiplier, and then by the
twelfth parts, setting the result of each term one place removed to
tbe right hand when the multiplier is inches, and two places when the
parts become the multiplier. The sum of these successive products
will be the answer reqmred.
Or, instead of multiplying by the inches, &c., take such parts of
the multiplicand as these are of a foot.
28
P0WBB8 AND ROOTS.
[PABT 1.
Examples.
1. Multiply \2ft. 7i ins. by 1 /t. 3 J tVw.
//. in$. '
12 7 4 or,
7 3 9
88 3 4
3 1 10
9 5 6
92 2 7 6
tnM.
3 == i of 1 ft.
9' = I of 3 ins.
fL %n», '
12 7 4
7 ft.
88 3 4
3 I 10
9 5^
92 2 7
2. Multiply Z5fi. ^\ins. into 12^. S^ins.
ft. ins. '
35 4 6 or,
12 3 4
tnM.
3 = ^ of 1 ft.
4' = ^ of 3 ins.
424 6
8 10 1
11 9
6
6
434 3 11
ft. ins. '
35 4 6
12 ft.
424
6
8
10
n
11
9i
434
3
_n
The feet in the answers are square feet, but the numbers standing
in the place of inches are not square inches but twelfth parts of
square feet, each part being equal to 1 2 square inches ; and the
numbers in the third place being twelfth parts of these are square
inches : in like manner, if the operation be carried further, every
successive place will be a twelfth part of that preceding it.
Sect. IX. Powers and Roots.
• A power is a quantity produced by mttltipl}'ing any given number,
called the root or radix, a certain number of times successively by
itself. The operation of thus raising powers is called involution.
Thus, if 3 is the root,
3 = 3 is the Ist power of 3.
3x3=3'= 9, is the 2d power, or square of 3.
3 X 3 X 3 = 3 '= 27, is the 3d power, or cube of 3.
3 X 3 X 3 X 3 = 3^ = 81, do. 4th power, or biquadraU of 3,
&c. &c. &c.
CHAP. I.]
BVOLUTION.
29
L ,.,
Table of thejirsi Nine Patters of the first Nine Numbers.
lit
A
2d
1
4
9
16
3d
1
8
4di
&th
6tk
7tli
&th
dUi
1
1
1
1
1
1
16
81
32
64
128
356
61t
243
729
2187
6fi61
65536
19683
64
G2&
1296
1024
3125
4096
1S625
16384
£^144
3S 1 1S5
78126
279936
390625
1679616
1953125
m
64
216
7776
4G6&G
10077696
343 240]
512 4006
16B07
117649
823543
2097152
5764801
— ^ —
16777216
4035364^7
32768
262144
134317728
tl
729 6561
^9049
531441
47629C^
43046731
367420480
So again, J x f = $ = square off;x = ^ = cube of
3 > ^ X J = ^, biqoadrate of 4 ; and so of others. Where it is
eyident, that while the powers of integers become successively larger
^nd larger, the powers of pure or proper fractions become sue
oessively smaller and smaller.
EVOLUTION.
Evdulumy or the extraction of roots, is the reverse of involution.
Any power of a given number may be found exactly ; but we
cannot, conversely, find every root of a given number exactly*.
Thus, we know the square root of 4 exactly, being 2 ; but we cannot
assign exactly the cube root of 4. So, again, though we know the
cttbe root of 8, viz. 2, we cannot exactly assign the square root of 8.
But, of 64 we can assign both the square root and the cube root, the
former being 8, the latter 4.
By means of decimals we can in all cases approximate to the root
to any proposed degree of exactness.
Those roots which only approximate are called surd roots, or surdSy
or irrational numbers ; as v^2, V5, \/9> &c., while those which can
be found exactly are called rational; asx/9 = 3, i/125 = 5,
iyie == 2.
1. — To extract the square root.
Rule, — Divide the given number into periods of two figures each,
by setting a point avei' the place of unitSy another over the place of
* For the method of extracting roots by logarithm!, see page 47*
dO EVOLUTION. [pari I.
hundreds, and so on over every second figure, both to the left hand in
integers, and to the right hand in decimals. Find the greatest square
in the first period on the left hand, and set its root on the right hand
of the given number, after the manner of a quotient figure in division.
Subtract the square thus found from the said period, and to the
remainder annex the two figures of the next following period, for a
dividend. Double the root abovementioned for a divisor ; and find
how often it is contained in the said dividend, exclusive of its right
hand figure ; and set that quotient figure both in the quotient aad
divisor. Multiply the whole augmented divisor by this last quotient
figure, and subtract the product from the said dividend, bringing down
to it the next period of the given number, for a new dividend.
Repeat the same process, viz., find another new divisor, by doubling
all the figures now found in the root; from which, and the last
dividend, find the next figure of the root as before; and so on through
all the periods, to the last*.
Note. — The best way of doubling the root, to form the new
divisors, is by adding the last figure always to the last divisor, as ap
pears in the following Examples. — Also, after the figures belonging to
the given number are all exhausted, the operation may be continued
into decimals at pleasure, by adding any number of periods of cyphers,
two in each period.
Examples.
1. Find the square root of 17*3056.
17*3056(4'16 the root : in which the number of
IG decimal places is the same as the
number of decimal periods into which
the given number was divided.
81
130
1
81
826
4956
6
4956
* The reason for separating the figures of the dividend into periods or
portions of two places each, is, that the square of any single figure never con
sists of more than two places ; the square of a number of two figures of not
more than four places, and so on. So that there will be as many figures in the
root as the g^iven number contains periods so divided or parted off.
And the reason of the several steps in the operation appears from the algebraic
form of the square of any number of terms, whether two or three, or more.
Thus, 36* » 30* + 2 . 30 . 5 + 6% or genmilly (a f 6)* = a* + 2 a 4  i» =
a* 4* (2 a 4* 6) ^t the square of two terms ; where it appears that a is the first
term of the root, and b the second term ; also a the first divisor, and the new
divisor is 2 a 4* 6, or double the first term increased by the second. And hence
the manner of extraction is as in the rule.
CHAP. I.]
EVOLUTION.
31
2. !Rnd the square root of 2,
to six decimals.
2(1414213 root.
1
24 1
4
00
96
281
1
400
281
2824
4
11900
11296
28^82
2
60400
56564,
28284
1 383600
1 282841
2828423
10075900
8485269
1590631
3. Find the square root of f^.
r\ = '4>16666666y &c.
64i 6666(064549, &c.
36
124 566
4 496
1285
5
7066
6425
12904 64166
4 51616
12908
9
1255066
1161801
93265
^ote. — In cases where the square roots of all the integers up to
1000 are tabulated, such an example as the above may be done more
easily by a little reduction. ThusV^j =\/(t^^ X ^f) =x/ j^^ =
^^60 = ?:?^ = 645497, &c.
2. — To extract cube and higher rooU.
The rules usually given in books of arithmetic for the cube and
higher roots, are very tedious in practice: on which account it is
advisable to work either by means of approximating rules, or by
means of logarithms*. The latter is, generally speaking, the best
method. We shall merely present here Dr. Hutton's approximating
rale for the cube root.
Rule, — By trials take the nearest rational cube to the given num
ber, whether it be greater or less, and call it the assumed cube.
Then say, by the Rule of Three, as the sum of the given number
and double the assumed cube, is to the sum of the assumed cube and
double the given number, so is the root of the assumed cube, to the
root required, nearly. Or, as the first sum is to the difference of the
given and assumed cube, so is the assumed root, to the difference of
the roots, nearly.
• See page 47
32 PROPOBTION. [part I.
Again, by using, in like manner, the cube of tbe root last found
as a new assumed cube, another root will be obtained still nearer.
And so on as far as we please ; using always the cube of the last
found root, for the assumed cube.
Example,
To find the cube root of 210358.
Here we soon find that the root lies between 20 and 30, and then
between 27 and 28. Taking therefore 27, its cube is 19683, which
is the assumed cube. Then
19683
2
210358
2
39366
210358
420716
19683
As 604018
: 617546
27
4322822
1235092
27 : 276047
604018)16673742(276047 the root nearly.
459338 Again, assuming 27*6
36525 and working as before, the
284 root will be found to be
42 2760491.
Sect. X. Proportion.
Two magnitudes may be compared under two different points of
view, that is to say, either by inquiring what is the excess of one above
the other, or hoto often one is contained in the other. The result of
this comparison is obtained by subtraction in the first case, by division
in the second, the quotient resulting being termed the ratio of the
two numbers. Thus 3 maybe regarded as the ratio of 12 to 4, since
*3p or 3 is the quotient of the numbers 12 and 4.
The first of two numbers constituting a ratio is called the antece
dent^ the second the consequent.
The difference of two numbers is not changed by adding one and
the same number to each, or by subtracting the same number from
each.
Thus 12  5 = (12 + 2)  (5 + 2) = 14  7 = (12  2) 
(5  2) = 10 — 3.
In like manner, a ra^io is not changed by either muUiplging both
its terms, or dividing both its terms by the same number.
Thus V =(y X !) = !« =c/f)=f
CHIP. I.] PROPOBTION. 33
F/lnaliiy of differences^ or equidifference^ is a term used to indicate
tlattbeMerence between two numbers is the same as the difference
between two other numbers. Such, for example, asl2 — 9 = 8 —
£gva/i<jf of ratios^ or proportion^ is similarly employed to denote
that tbe ratio of t^wo numbers is the same as that between two others.
TbusSO and 10, 14 and 7, have 2 for the measure of the ratio : we
kie therefore a proportion between 20 and 10, 14 and 7, which is
tbos expressed, 20 : 10 : : 14 : 7, and thus read 20 are to 10 a« 14
are to 7. The same proportion may also be represented thus, ^g = y .
Tboagb, \>y whatever notation it be represented, it is best to read or
eoamer&te it as above. It is true, however, that in all cases when
tvo fractions are equal, the numerator of one of them is to its de
QOffiinator, as the numerator of the other is to its denominator.
In a proportion, as 20 : 10 : : 14 : 7, the second and third terms
we called the meanSy the first and fourth the extremes.
Wben the two means are equal, the proportion is said to be con
tinued. Thus 3 : 6 : : 6 : 1 2 are in continued proportion. This is
osually expressed thus rr 3 : 6 : 1 2 ; and the second term is called
the mean proportional.
In the case of equidifference^ as 1 2 — 9 =7 — 4, the sum of the
extremes (12 + 4) is equal to that of the means (9 r 7). In like
JMnnerin a proportion, as 20 : 10 :: 14 : 7, the product of the ex
tremes (20 X 7) is equal to that of the means (10 x 14). The
converse of this likewise obtains, that if 20 x 7 = 10 X 14, then
20:10:: 14 : 7. Hence,
1. If there be four numbers, 5, 3, 15, 9, such that the products
5 X 9 and 3 x 15 are found equal, we may infer the equality of
tbcir ratios, or the proportion ^ = ^ , or 5 : 3 : : 15 : 9. So that a
proportion may always be constituted with the factors of two equal
products.
2. If the means are equal, their product becomes a square ; there
fore the mean proportional between two numbers is equal to the square
root of their product. Thus, between 4 and 9 the mean proportional
i8%/(4 X 9) = 6.
3. If a proportion contain an unknown term, such, for example, as
5:3:: 15: the unknown quantity; since 5 times the unknown
<IttDtity roust be equal to 3 x 15 or 45, that quantity itself is equal
to 43 r 5 or 9. Or generally, one of the extremes is equal to the
prodact of the means divided by the other extreme ; and one of the
OKans is equal to the product of the extremes divided by the other mean.
4. We may, without affecting the correctness of a proportion,
object the several terms which compose it to all the changes which
ciQ be made, while the product of the extremes remains equal to that
ofthemeans. Thu8,for5 : 3 : : 15 : 9, which gives 5 x 9 = 3 x 15,
we may
I. Change the places of the means without changing those of the
^^Etremea, or change the places of the extremes without changing those
of the means : this is denoted by the term aUernando,
D
84 BULK OF THREE. [^A&T I.
Thus, 5 : 3 : : 15 : 9
become 5 : 15 : : 3:9
or 9 : 3 : : 15 : 5
or 9 : 15 :: 3 : 5
II. Put the extremes in the places of the means ; this is oalled
invertendo ; as
3 : 5 : : 9 : 15
III. Multiply or divide the two antecedents or the two consequents
by the same number.
It also appears, with regard to proportions, that the sum or the dif
ference of the antecedents is to that of the consequents, as either ante
cedent is to its consequent.
And, that the sum of the antecedents is to their difference, as the
sum of the consequents is to their difference.
Tu ^ 3; 15 , ,, , 5 h 15 5^15
^^"^ "3"T^ = 1^ = v> and ^^ —rz^'
If there be a series of equal ratios represented by ^ = ijp = y ==
6 ^ 10 + 14 f 30
3g, we shall have 3 ^ ^ ^ ^ ^. ^^ = JJ =  = V> = &c.
Therefore, in a series of equal ratios, the sum of the antecedents is to
the sum of the consequents, as any one antecedent is to its consequent.
If there be two proportions, as 30 : 15 : : 6 : 3, and 2 : 3 : : 4 : 6,
then multiplying them term by term, we shall have 30 x 2:15 x
3 : : 6 x 4:3 x 6, which is evidently a proportion, because 30 X
2x3x6 = 15x3x6x4 = 1080.
Thus, also, any powers of quantities in proportion are in proportion;
and conversely of the roots. Thus,
If 2 : 3 : : 6 : 9 then 2* : 3' : : 6'* : 9'
2 : 3 : : 6 : 9 „ \/2 : ^3 : : v/6 : v 9
2 : 3 : : 6 : 9 „ 2^ : 3' : : 6^ : 9'
RULE OP THREE.
When the elements of a problem may be so disposed that they
form a proportion of which the quantity sought is the last term ; that
is, when the first bears the same proportion to the second as the third
does to the fourth or unknown quantity, its value may be easily
determine<l, and the problem is said to belong to the Goklen EuUj or
Bute of Three.
Etdle. — Of the three given terms set down that which is of the
same kind as the number sought, then consider from the nature of the
problem whether this number will be greater or less than the term
so put down ; if greater^ write on its right hand the greater, if leu^
the lesser, of the two remaining terms, and place the other on its left
hand.
Then multiply the second and third terms together, and divide their
product by the first; the quotient will be the number sought It
CEAP. I.]
BULB OF THR£B.
85
moat be observed, that the first and third terms must be reduced to
tlie nine deDomiiiation ; and if the second term is a compound
nomber, it should be reduced to the lowest name mentioned ; unless
the ibird term is a composite number, in which case it is generally
better to multiply the second term (without any previous reduction)
h the componeDt parts of the third, as in compound multiplication,
ifler which divide the compound product by the first term, or by its
&ctoi8. The answer will be of the same denomination as the second
tern.
Examples,
1. If 3 gallouB of brandy cost 2. How much brandy may be
19i.,what will 126 gallons cost bought for 39/. 18«., at the rate of
It the same rate ? { 3 gallons for 1 9 shillings ?
fA, t. gal. !
3:19:: 126 : ?
19
1134
126
3)2394(798 sbillings
21
— or 39/. 18«. Ans.
29 '
27
24
24
3. If 21 yards of cloth cost
2*^ 10#., what will 1 60 yards cost ?
frfi. £ «. yds.
Hefe,21:24 10:: 160 : ?
4
4x4x 10=160
98
4
19
gaL
: 3 :
18
?
21
392
10
(3)3920
£
39
20
798*.
3
19) 2394 ( 126 ^wj.
49
38
114
114
4. If by selling cloth at 1/. 2«.
per yard, 10 per cent, is gained,
what would be gained if it had
been sold at 1/. 5b. per yard ?
£ t. £ 9.
Here, 1 2: 110:: I 5 : ?
20 20
22
25
no
2750
22
[7)1306 13 4
£186 13<. 4d. Ana.
(2)
111 )1375
Amount £ 1 25
Deduct 100
Gain per cent. £ 25
= d2
3^
aru :f thru.
[PABT I.
of 5'».'. ::r 5 Te^rsw s: 4
£ ^' £
Here. H» : 4 : : o^M
4
l<Ni 2244?
per
^. If 100 workmen can finish
a piece of work in 1 2 days, how
manj men working equally hard
would hare finbhed it in 3 davs ?
</. ir. d.
12: 100:: 3 : ?
12
3)1200
Aftneer 400 workmen.
£224
2*1
80
Interest for I vear, £22 $#
Then 1 : 22 S : : 5
5
;fll2 rt^u^rvr.
A distinct rule is usually given for the working of problems in
Compound Frojxtrtion ; but they may generally be solved with
greater mental facility by means of separate statings. Thus : —
7. If a person travel 300 miles
in 10 days of 12 hours each, in
how many days of 1 6* hours each
may he travel 600 miles ?
First, if the days ^ ere of the
same lenj^h, it would be, bv
simple proportion,
m. d, m.
As 300 : 10 : : 600 : 20 days.
But these would be days of 12
hours each, instead of 16, of
which fewer will be required.
Hence, again, by simple propor
tion,
h. d. h, d.
As 12 : 20 :: 16 : 15
So that the answer is 1 5 days.
8. If a family of 9 persons
spend 480/. in 8 months, how
much will serve a family (living
upon the same scale) of 24 per
sons 16 months?
I First, as 9 : 480
P
24:o£l280.
But this would only be the
expense for 8 months. Hence,
again.
m. £
As 8 : 1280
171. £
As 8 : 1280 : : 16 : 2560, tbe
expense of the 24 persons for 1 6
months.
yote. — The Rule of Three receives its application in questions of
Interest^ Discount^ Fellowskipy Bartery &c.
CHAP. I.]
BBTXRMINATION OF RATIOS.
37
BBTERMINATION OP RATIOS.
To find the ratio of two numberSy A and B, to each other.
Rule 1. — DiTide JB hy A, then, unit^ or 1 : the quotient, will be
tbe iim ratio ; and if the quotient be an integral number, the ratio
will be expressed in the least terms possible. If, however, it contain
a fiMOoD, proceed as follows : — first write, 1 : integral portion of
Ike quUient f the numerator of the fraction ; then, add 1 to the
integral, and subtract the numerator of the fraction from the de
nominator, and write under the former, 1 : integral portion 41 —
tk 's.ofihe numerator and denominator.
Then, if the numerator and this difference bo nearly equal, add
the two ratios together ; but if the numerator and difference are not
nearly equal, divide the greater by the less ; then, multiply that ratio
ending with either the numerator or the difference, whichever was
the divisor, by the integral portion of this last quotient, and add to it
the other ratio. With the three ratios thus obtained proceed in the
same manner, from whichever two have the numbers appended by
the signs  and — , nearest equal, to obtain a fourth ratio, and thus
continue, until this appended number has been eliminated.
Note. — The ratios thus found will be alternately greater and less
than the true one, but continually approaching nearer to it. And
that is the nearest in small numbers, which is immediately followed
bj much larger numbers : the excess or defect of any one is equal
to a fraction, having the number appended to that ratio by the sign
+ or — for its numerator, and the denominator belonging to the
first quotieDt, for its denominator.
Example \,
To find the ratio of 10000 {A) to 7854 {B) in small numbers.
7854 H 10000 =0t7^«5^
Then,
Ist
I.
2Dd
3rd
II.
4th
III.
5tb
IV.
6th
V.
7th
8th
VI.
8th
VII.
10th
VIII.
nth
1 : + 7854 or ratio of 1 to the integer and num.
1 : 1—2146 or integer 41— the '^ of num. and den.
2146)7854(3
d : 8—6438 or 2nd ratio x by 3.
4: 3 + 1416 or 1st and 3rd ratios added together.
5 : 4— 730 or 2nd „ 4th „ „
9 : 7+ 686 or 4th „ 5th „ „
14: 11— 44 or 5th „ 6th „ „
44)686(15
210 : 165— 660 or 7th ratio x by 15.
219 : 172+ 26 or 6th and 8th ratios added together.
233: 183— 18 or 7th „ 9th „
452: 355+ 8 or 9th „ 10th „ „
8)18(2
38
DETERMINATION OP RATIOS.
[PABT
IX.
X.
12th
Idth
Uth
15th
904 : 710f 16 or llth ratio x by 2.
1137 : 893— 2 or 1 0th and 1 2th ratios added together.
2)8(4
4548 : 3572— 8 or 13th ratio x by 4.
5000 : 3927 f or llth and 14th ratios added together.
The ratios are numbered according to their convergence, with
Roman numerals to the left hand, and are as follows : —
± A Ji
17 3 > T)
A ±Jl AX9 AAA AAJL
7> 1I> 112 9 183> 366>
OJLl 5000
853 > 3 92 T '
Of these the nearest in small numbers is J^, differing from the
true ratio by only j^^QQt and is indicated (as alluded to in the
preceding note) by being immediately followed by the much larger
numbers f^^.
Example 2.
To find the ratio of 268*8 to 282 in the least numbers.
2688) 2820 (l,VftS
2688
Then,
I.
l8t
2nd
II.
3rd
4th
III.
IV.
5th
eth
7th
V.
8th
9tb
1
I
19
20
40
41
61
183
224
132
: 1 h 132 or ratio of 1 to the integer and num.
: 2—2556 or integer  1— the '^ of num. and den.
132)2556(19
: 19 + 2508 or 1st ratio x by 19.
: 21 — 48 or 2nd and 3rd ratios added together.
48)132(2
: 42— 96 or 4th ratio x by 2.
: 43+ 36 or 1st and 5 th ratios added together.
1 2 or 4th „ 6th ,, „
: 64—
12)36(3
: 192—
: 235
36 or 7th ratio x by 3.
or 6th and 8th ratios added together.
Therefore the several ratios are J, ?% J^, J, and j^. And
the excess or defect of any one is seen by inspection; thus, *^ differs
from the true ratio only jj^; and ^, but ^11^.
Rule 2. — Divide the greater number by the less, and the divisor
by the remainder, and the last divisor by the last remainder, and so
on till remain. Then,
1 divided by the first quotient, gives the first ratio :
And the terms of the first ratio multiplied by the second quotient,
and 1 added to the denominator, give the second ratio :
And in general the terms of any ratio, multiplied by the next quo
tient, and the terms of the foregoing ratio added, give the next suc
ceeding ratio.
CHA?.I.] DBTSIUCINATION OV &ATIOe* 39
Example 3.
LetthenoTObcrs be 10000 and 31416, or the ratio Hf?#
10000)31416(3
30000
1416)10000(7
9913
88)1416(16
88
536
528
Tben,
8)88(11
88
< s= Ist or least ratio.
17 7 7
i Y 7 — and = — = 2nd ratio.
8 ^ 21 21 ^ 1 22
7 112 , 112 + 1 113 Q , ,.
__Y ift— and • — = s=3rd ratio.
22''^^'" 862 352 43 355
lL^n'^^ and i?l£±:L = l?^=4tli ratio.
S65 "^ ^^ " 3905 3905 + 22 3927
JEfzample 4.
The mtio of 268*8 to 282 is required.
2688)2820(1
2688
132)2688(20
264
48)132(2
96
86)48(1
86
12)36(3
36
40 LOGARITHMIC ARITHMETIC. [PABT
Then,
 := 1st ratio.
 X 20= and ^^^^ = ^^ = 2nd ratio.
20 ^ 40 , 40 4 1 41 « , .
21 ^ ^=42^"^^ IT^l = 43 =^r<l^
41 , 41 ,41+20 61 , ^ .
■TT X 1 = — and 7 —  =  = 4th ratio.
43 43 43 + 21 64
61 „ 183 ^ 183 +41 224 ,^ .
gj X 3 = — and j^^^^ = — = 5th ratio.
Sect. XI. Logarithmic Arithmetic.
As the nature and properties of logarithms are described in a sub
sequent part of this work, being so placed because such description
could not have been properly understood without a certain acquaint
ance with algebra, we shall here only explain the use of the tables
given in the Appendix, and the method of employing logarithms to
facilitate the common processes of arithmetic.
By an inspection of Tabic II., which contains the logarithms of all
numbers from 1 to 100, it will be seen that each logarithm consists
of two distinct parts, separated by a decimal point ; thus, the loga
rithm of 13 is 1*113943; the number to the left of the decimal
point (or 1 in the above example), is called the index or charac
teristic*^ and its value depends only upon the number of digits in the
quantity whose logarithm it is, without any regard to thet7a/tt€ of that
quantity, and it is always 1 less than that number of digits ; thus, in
the example, the characteristic of the logarithm of 13, which contains
two digits, is 1, or one less than that number ; and it will be seen from
the Table, that 1 is the characteristic of all the logarithms from 10
to 99, but that, for numbers below 10, the index is 0, and for 100
is 2, in each case 1 less than the number of digits in the quantity of
whose logarithm it is the characteristic. The characteristic, there
fore, of the logarithms of all numbers
equal to or greater than 1 and less than 10 is 0*
10
»
100 „ 1
100
>»
1000 „ 2
1000
>»
10000 „ 3
10000
»>
100000 „ 4
&c.
&c. &c.
* In order to avoid confusion from the use of the leord index to signify two
thincB, we shall throughout this work employ the term characterisHc when
speaking of logarithms, and iruies when speaking of roots or powers.
CHAP. I.] LOGARITHMIC ARITHMETIC. 41
When the qaantity is less than unity, the characteristic of its loga
rithm becomes negative, and its value is determined hy the number
of cyphers which occur between the decimal point and the first signi
ficant figure, (the fraction being decimally expressed), and is always
1 greater than such n amber of cyphers; or it is equal to the differ
ence in the number of figures in the numerator and decimal denomi
nator; thas, the characteristic of the logarithm of
•1
or
iV
is
1^
•01
»
lio
»»
2
•001
»
16^6
>»
3
0001
&c.
>»
1
?»
4.
10000
&c.
&c.
The decimal part of the logarithm, or that lying to the right of
the decimal point, depends entirely on the relative value of the figures
composing the quantity whose logarithm it is, and not at all upon the
tttoal numerical value of that quantity ; thus, in the example already
giTen, the decimal part of the logarithm of 13 is '113943, which is
iko Ae decimal part of the logarithm of 1*3, or 130, or 1300, for in
each case the 1 and the 3 have the same relative value. So that the
decimal portion of a logarithm is always the same for the same
fignrea, and is not altered by the addition of any number of cyphers
either to the right or to the left hand of those figures, or what is
equivalent, by the multiplication or division of the quantity by 10,
or any power of 10 ; it is only the characteristic of the logarithm
which alters its value, 1 being akded to the characteristic for every 10
hy which the quantity is mvltipliedy or subtracted from it for every 10
hy which the quantity is (/«t7M/e(/. Thus,
the logarithm of 745800
being 5872622
that of 74580
is
4872622
„ 7458
>»
3872G22
7458
>>
2872622
7458
»>
1872622
7458
»
0*872622
•7458
99
1872622
„ 07458
>»
2872622
•00745fi
^ »
3872622
It must be borne in mind, that in the logarithm of a fractional
qnantity, it is only the characteristic which has a negative value, and
that the decimal pal^t of a logarithm is always positive. It is, how
CTer, sometimes convenient to have the whole logarithm expressed
negatively, both characteristic and decimal ; for which purpose, sub
• The negative sign ( — ) ia always placed above the characteriatic, thus 2,
initcad of before it, in order to avoid its being misunderstood for the sign of
mbcractiim.
42 LOGARITHMIC ARITHMETIC. [PART I.
tract the last right hand figure in the decimal portion from 10, and
all the others from 9, and the result will he what is termed the
arithmetical complement of the decimal, to which prefix the former
characteristic less I, and the result will he a negative logarithm,
equivalent in value to the original logarithm having only a negative
characteristic; for example, the logarithm of '07458, as above, is
2^872622, which is equivalent to — 1 127378. It is also frequently
convenient to take the arithmetical complement of the whole
logarithm, and this is obtained by subtracting the right hand figure
of the decimal from 10, and all the others from 9, including the
characteristic when positive^ but if negative it must be added to 9.
Thus, the arithmetical complement
of 314G128 is 6853872
„ 207G276 „ 11923714
„ 5322839 „ 4677161
„ i986772 „ 10013228
USB OP THE TABLES.
To find the logarithm of any given number.
If the number is less than 100, its logarithm will be found in
Table II., with its proper characteristic prefixed ; but if the number
contains more than two figures, its logarithm may be found from
Table III, as follows: — If there are only three figures in the num
ber, look for that number in the first column of the table, and on the
same line in the next column to the right, under 0, will be found the
decimal portion of the required logarithm, to which the proper cha
racteristic must be prefixed, according to the rules which we have
just explained. If the quantity contains four figures, look for the
first three figures in the first column as before, and the four last
figures of the logarithm of the required number will be found on the
same line with those three figures, and in that column which has at
its head the fourth figure of the given number ; the two first figures
of the logarithm will be found in the second column (headed 0), and
which figures being common to all the logarithms enclosed by each
pair of horizontal lines, it is unnecessary to repeat. Where these
first figures change their value in the middle of a line, the same
is indicated by a break in the horizontal line, thus, 139879  0194,
which shows that the two first figures (13) have changed to 14, and
the right hand logarithm is therefore 140194. The heading figures are
repeated at every tenth line in the body of the tables, in order to
facilitate their use.
Examples.
Required the logarithm of 734.
In Table III., on the same line with 734 and under 0,. are found
5696, the four last figures of the logarithm^ to which the common
CHIP. I.] LOOARtTHMIC ABITBlfBTIC. 48
figures 86 and the proper characteristic 2 being prefixed, we obtain
2'^569C, the logarithm required.
Find the logarithm of 3476.
Hpre, on the same line with 347 and under 6, will be found 1080,
which, with the two first figures and the characteristic prefixed, is
3'54108O, the logarithm required.
The log. of 584 is ^766413
„ 0932 „ 2969416
1024 „ 1 010300
„ 3708 „ 3569140.
When the quantity whose logarithm is required contains more than
four figures, proceed as follows: — Find the logarithm for the first four
figures as above, then look in the first column of Table I. for the first
figures, and on the same line in the column having at its head the
fifth figure will be found the quantity which must bo added to the
logarithm already taken out^ to give the logarithm of the quantity first
required. If the first four figures are not found in the first column
of the table, then take the line containing the next less number to it.
If the number whose logarithm is required contains more than ^yc
figures, proceed as above to obtain the logarithm of the first five
figures, then, on the same line of Table I. that the number added to
the logarithm for the fifth figure was found, and in the column
htTJDg at its head the sixth figure, will be found a quantity, which,
dirided by 10 (or what is the same, having its right hand fisure taken
twtj*), and added to the logarithm already found, will give the
iogtfithm of the first six figures; again, on the same line and in the
cokmn baring at its head the seventh figure, will be (bund a quan
tit?, which, divided by 100 (or baring two figures cut off from the
right band), and added, will give the logarithm for seven figures t.
Examples.
Required the logarithms of 11488, 621547, 768654, 7642179.
log. of the first four figures from Tab. Ill = 4059942
From Tab. I. on line with 1148 and under 8 ... = 302
Logarithm of 11488, as required = 4060244
Log.of6215 =1= 5793441
From Tab. I. on line with 6160, the next less) _ „«
No. in the tab. to 62)5 and under 4 J ""
On aame line under 7 = 4
Logarithm of 621547 = 5793474
* If the figure thus cot off ezoeeds five, one must be added to the first right
kind fijnire left.
t See remark at page 45, with regard to the number of places to be de
44 LOOABITHMIC ARITHMETIC. [PART I.
Log. of 7686 = 5885700
From Tab. I. on same line with 7686 under 5... = 28
On same line under 4 := 2
Logarithm of 768654 = 58 85730
Log. of 7642 = 3883207
From Tab. I. on same line with 7552 under 1 ... = 5
On same line under 7 = 3
On same line under 9 =
7
99
513
Logarithm of 76421 79 = 3883217
To find the number answering to any given logarithm.
Look in Table III. for the given logarithm, or the next less in
value to it that can be found, then on the same line, in the first
column, will be found the first three figures, and at the head of the
column in which the logarithm was found, the fourth figure of the
number sought. If the given logarithm is found exactly in the table,
the figures thus obtained will be the required number, care being
taken to point off one more figure to the left hand than there are
units in the characteristic of the given logarithm, cyphers being at
tached to the right hand of the number, if requisite. If, however,
the given logarithm is not found exactly in the table, subtract from it
the next less logarithm found, calling the remainder the first differ
ence ; then look in Table I. on the same line with the four figures
already obtained from Table III. (or the next less figures which can
be found) for this difference, and at the head of the column in which
it is found will be the fifth figure of the number sought. If the first
difference is not found exactly in the table, look for the next less
number to it, which subtract from the first difference for the second
difference ; then add a cypher to this second difference, and look for
it on the same line of Table I. as before, and the figure at the head
of the column containing the nearest number to it, either greater or
less, will be the sixth figure of the number required.
Examples.
Required the number answering to the logarithm 3*241756.
Given log. = 3'241 756
Next less log. in Tab. III. =3*241546 = the log. of 1744*
210 first dif.
In Tab. I. on same line with ) i nn • r j • i o o
the next less No. to 1 7*4 2^ '^ *^"""^ '" *="'• * ]^
110 second dif.
On the same line 99 is found in col. 4 *04
The No. required = 1744*84
CHAP. 1.] LOGARITHMIC ARITHMETIC. 45
In this example the next less logarithm which can he found in
Table III. is 3*241546, the numher answering to which, 1744, is the
first four figures of the namher sought ; then subtracting this loga
rithm from the given logarithm, we obtain for the first difference
210, and looking in Table I. on a line with 1740 (the next less
number to 1744), for the next less number to 210, we find 199, at
tbe head of the column containing which is 8, the fifth figure re
quired; then subtracting 199 from 210, we obtain the second differ
ence, 11, and adding a cjrpher, the nearest number which we find on
the same line is 99, at the head of the column containing which is 4,
the sixth figure required.
Required the numbers answering to the following logarithms : —
3510009, 2475771, 5871624.
The number answering to the logarithm 3*510009 is found at
oQcetobe32d6.
Given log. = 2476771
Next less log. = 2475671 = the log. of 2990
From Tab. I....
100 = 1st dif.
87
•06
130 = 2nd dif.
130
009
No. required =
■
= 5871624
= 5871573 = the log. of
299069
Given log.
Next less log.
744000'
Prom Tab. I....
51 = 1st dif.
46
80
60
62
9
The No. required = 744089
It should be observed here, that the number of figures which may
^ depended upon in any result obtained by logarithms, will be equal
Jo the number of decimal places in the logarithms employed ; thus,
in Qiing the tables appended to this work, the results obtained will
^ accurate to six figures, except towards the end of the tables, in
which only five figures should be trusted.
MULTIPLICATION AND DIVISION BY LOGARITHMS.
To mulHpiy two numbers together, add together their logarithms,
4i6 LOOABITHIfIC ARITHMETIC. [PART I.
and the sum will be the logarithm of their product; or, to divide
one number by another, subtract the logarithm of the divisor from
the logarithm of the dividend, and the remainder will be the log
arithm of the quotient of the two numbers.
Ex, — Multiply 80 X 43 X 72*64; and divide the product of
(7143 X 6278) by 3145.
Add
Log. 80 = 147712J; Log. 7143
log. 43 = 1633468 add log. 6278
Jog. 7254 = 1860578
Log. 935764 = 4971167
sub. log. 3145
: 3'85S881
3797821
7651702
2497621
Log. 142589 ;= 5154081
rROPORTION OR THE RULE OP THRBB BY LOGARITHMS.
The Rule of Three is very readily performed with the aid of log
arithms, by simply adding together the logarithms of the 2nd and
3rd terms, and subtracting the logarithm of the 1st, the remainder
being the logarithm of the 4th term, or number required to be
found. Or, instead of subtracting the logarithm of the 1st term, we
may €idd its complement, (the method of obtaining which has been
already explained at page 42,) and subtract 10 from the character
istic of the result, which will, as before, be the logarithm of the
4th term.
Examples.
The following are the same as the 1st, 3rd, 4th, and 5th Ex
amples given in the Rule of Three, at pages 35 and 36.
1st Ex.
Log. 19 = 1278754
+ log. 126 = 2100370
3379124
 log. 3 = 04771 21
3rd Ex.
Log. 245 = 1389166
4 log. 160 = 2204120
3593286
 log. 21 = 1322219
Log. 798 =s 2902003 Log. 186667 = 2271067
4th Ex. 5th Ex.
Comp. of log. 22 = 8657577 Comp. of log. 100 = 8000000
f log. 110 = 2041393 + log. 4 = 0602060
+ log. 25 = 1397940 + log. 560 = 2748188
Log. 125 = 2096910
Log. 224 = 1350248
CHIP. I.J LOGABITHMIC ARITHlfKTIC. 47
EVOLUTION AND INVOLUTION BY LOOABITHM8.
To perfonn the operation of involution^ or the raising of powers,
it is only necessary to multiply the number, any power of which is
required, by the index of that power, and the product will be the
logirithm of the required power; and, inversely, the operation of
miuiioH^ or the extraction of roots, is performed by simply dividing
the logarithm of the number by the index of the root required, the
quotient ynl\ be the logarithm of the root
Examples.
Square 84, cube 13, and raise 7 to the sixth power.
Log. 84 = 1924279 x 2 = 3848558 = 7056 = 84
log. 13 = 11 13943 X 3 = 3341829 = 2197 = 13»
log. 7 = 0846098 x 6 = 5070588 = 117649 = 7^
Extract the square root of 576, the cube root of 4913, and the
axth root of 46656.
Log. 576 = 2760422 r 2 = 1 3802 11 = 24 = n/ 576
log. 4913 = 3691347 ^ 3 = 1230449 = 17 = 'V 4913
log. 46656 = 4668908 r 6 = 0779151 = 6 = V46666 •
It is necessary here to make a few remarks on performing the
operations of evolution and involution on logarithms with negative
cWacteristics. In doing this, it must be borne in mind that it is
only the characteristic which has a negative value, the decimal part of
tie logarithm being always positive ; therefore, if it is required to
multiply a logarithm with a negative characteristic by any number,
iim multiply the decimal part of the logarithm, pointing oflf as many
decimal figures in the product as there were in the logarithm, then
multiply the characteristic, and subtract from the product the num
ber (if any) pointed off to the left in the first product, the result will
W the negative characteristic, and the decimals pointed off in the
first product will be the decimal part of the required logarithm.
of
Examplei.
Required the square of *25, the cube of '375, and the sixth power
The logarithm of 25 = T397904
First multiply 897940 by 2
2
_ _795880
Then 1x2=2
2795880 = 0625 = 25*.
48 LOGARITHMIC ARITHMETIC. [PART I.
The logarithm of 375 is f574031.
•574031
3
1722093
1 X 3 = 3
2722093 = 05273437 = •375».
The logarithm of 7 is 1845098
845098
6
5070588
1x6 = 6
1070588 =117649 = •7«.
To divide a logarithm with a negative characteristic by any nam
ber :— If the characteristic is a multiple of that number, that is, if k
is divisible by it without remainder, proceed as in ordinary division ;
if not, separate the characteristic from the decimal, and add to the
characteritttic a number which will make it divisible^ and prefix to the
decimal the same number, then divide both by the given divisor,
and the quotients will be the characteristic and decimal of the
logarithm required.
Examples,
Required the square root of 0625, the cube root of 74, and the
fifth root of 543.
The logarithm of 06*25 is 2795880
Then 2)2795880
r397940 = 26 = \/0625
The logarithm of 74 =1869232.
Then 1+2=373 = 1
and 2860232 H 3 = 956411
V74 = 9045 = 1956411
The logarithm of 543 is 1734800
Then T f 4=5r5 =1
and 4734800 f 6 = 946960
V543 = 885034 = 1946960
CRIP. 1.] PBOPRRTIRS OF NUMBERS. 49
Sbct. XII. Properties of Number m.
To render these intelligible to the student, we shall here collect a
few definitions.
Def. I. A tintV, or unity ^ is the representation of any thing con
adered individoally, without regard to the parts of which it is com
2. An integer is either a unit or an assemblage of nnits ; and a
Jhetum is any part or parts of a unit.
3. A multiple of any number is that which contains it some exact
namber of times.
4. One number is said to measure another, when it divides it with
out leaving any remainder.
5. And if a number exactly divides two, or more numbers, it is
then called their common measure,
6. An even number, is that which can be halved, or divided into
tvo equal parts.
7. An odd number, is that which cannot be halved, or which differs
from an even number by unity.
8. A prime number, is that which can only be measured by 1 , or
onitT.
9. One number is said to be prime to another when unity is the
odv Dnmber by which they can both be measured.
10. A composite number, is that which can be measured by some
Dumber greater than unity.
11. A perfect number, is that which is equal to the sum of all its
difigore, or diquot parts : — thus 6= f + f + f •
Pnp. 1. The sum or difference of any two even numbers is an
tten number.
2. The sum or difference of any two odd numbers is even ; but the
nm of three odd numbers is odd,
3. The sum of any eifen number of odd numbers is even ; but the
nm of any odd number of odd numbers is odd,
4. The sum or difference of an even and an odd number is odd,
5. The product of any number of even numbers is even ; and any
power of an even number is even,
6. The product of any number of odd numbers is odd; and every
power of an odd number is odd,
7. The product of any number of even numbers, by any number of
<^nombers, is even.
8. An odd number cannot be divided by an even number, without
t remainder.
9. If an o</<ar number divides an even number, it will also divide the
Wfofit
10. If a number consist of many parts, and each of those parts
^n a common divisor d, then will the whole number, taken col
^▼ely, be divisible by d,
11. Neither the sum nor the difference of two fractions, which are
n their lowest terms, and of which the denominator of the one con
E
50 PR0PBRT1B8 OF NUUBRR8. [PART T.
tains a factor not common to the other, can be equal to an integral
number.
12. If a square number be either multiplied or diTided by a square,
the product or quotient is a square ; and conversely, if a square num
ber be either multiplied or divided by a number that is not a square,
the product or quotient is not a square.
1 3. The product arising from two different prime numbers cannot
be a square number.
] 4. The product of no two different numbers prime to each other
can make a square, unless each of those numbers be a square.
15. The square root of an integral number, that is not a complete
square, can neither be expressed by an integer nor by any rational
fraction.
IG. The cube root of an integer that is not a complete cube cannot
be expressed by either an integer or a rational fraction.
17. Every prime number greater than 2, is of one of the forms
4n 4 1, or 4» — 1.
18. Every prime number greater than 3, is of one of the forms
6n r 1, or G w — 1.
1 9. No algebraical formula can contain prime numbers only.
20. The number of prime numbers is infinite.
21. The first twentv prime numbers ore 1, 2, 3, 5, 7, 11, 13, 17,
19, 23, 29, 31, 37, 4f, 43, 47, 53, 59, 61, and 67.
22. A square number cannot terminate with an odd number of
cyphers.
23. If a square number terminate with a 4, the last figure but one
(towards the right hand) will be an even number.
24. If a square number terminate with 5, it will terminate with 25.
25. If a square number terminate with on odd digit, the last figure
but one will be eren ; and if it terminate with any even digit, except
4, the last figure but one will be odd,
2G. No square number can terminate with two equal digits, ex
cept two cyphers or twoybwr*.
27. No number whose last, or righthand digit is 2, 3, 7, or 8, is
a square number.
28. If a cube number be divisible by 7, it is also divisible by the
cube of 7.
29. The difference between any integral cube and its root is always
divisible by 6.
30. Neither the sum nor the difference of two cubes can be a
cube.
31. A cube number may end with any of the natural numbers
1, 2, 3, 4, 6, 6, 7, 8, 9 or 0.
32. If any series of numbers, beginning from 1, be in continued
geometrical proportion, the 3rd, 5th, 7th, &c. will be squares ; the
4th, 7th, 1 0th, &c. cubes ; and the 7th, of course, both a square and
a cube.
33. All the powers of any number that end with either 5 or 6,
will end with 6 or 6, respectively.
CHAP. 1.] PROPERTIES OF NUMBRRS. M
U. Any power, w, of the natural numbers, 1, '2, 3, 4, 5, 0, &c.
las as many orders of differences as there are units in the common
exponent of all the numbers; and tlie last of those differences
U a constant quantity, and equal to the continual product
Ix*2x3x4x xw, continued till the last factor,
or the number of factors be w, the exponent of the powers. Thus,
Tbe 1st powers I, Si, 3, 4, 5, &c., have but one order of
differences 1111 &c., and that difference is 1 .
The iind powers I, 4, 9, 16, 25, &c., have two orders of
(Ufferences 8 5 7 9
of which tbe last is constantly 2 = 1 x 2.
The 3rd powers 1, 8, 27, 04, 125, &c., have three orders of
differences 7 19 37 61
12 18 24
6 6
of which the last is 6 = 1 X 2 X 3.
In like manner, the 4th, or last, differences of the 4th powers,
•re each = 24 = 1 x 2 x 3 x 4 ; and the 5tb, or last differences
of the 5th powers, are each 125 =: 1 x 2 x 3 x 4 x 5.
35. If unity be divided into any two unequal parts, the sum of
the square of either of those parts added to the other is the same.
Thosj of tbe two parts ^ and ♦,  + (f )« = ^ h (\) =: ^ ; so,
•g«n, of the parts J and ^, ? + (f)« = J + {fY = if *
For tbe demonstrations of these and a variety of other properties
of nmnbers, those who wish to pursue this curious line of inquiry
Bttjconsalt Legendre "Sur la Theorie des Nombres," the "Dis
<iaifltioDet Arithmeticse" of Gauss, or Barlow's " Elementary Inves
tigition of the Theory of Numbers."
Alto, for the highly interesting properties of Circulating Decimals^
ttd their connexion with prime numbers^ consult the curious works
rf the late Mr. H. Goodwyn, entitled " A First Centenary," and " A
Ttble of the Circles arising from the Division of a Unit by all the
htesenfrom 1 to 1024."
K 2
52 DEFINITIONS [pART I.
CHAP. II.
ALGRBRA.
Skct. I. Dejinitunis and Notation.
Algebra is the science of the computation of magnitudes in gene
ral, as arithmetic is the particular science of the^ computation of
numhers.
Every figure or arithmetical character has a determinate and indi
vidual value ; the figure 5, for example, represents always one and the
same number, namely, the collection of 5 units, of an order depend
ing upon the position and use of the figure itself. Algebraical cha
racters, on the contrary, must be, in general, independent of all par
ticular signification, and proper to represent all sorts of nambers or
quantities, according to the nature of the questions to which we apply
them. They should, moreover, be simple and easy to trace, so as to
fatigue neither the attention nor the memory. These advantages are
obtained by employing the letters of the alphabet, a, A, c, &c. to
represent any kinds of magnitudes which become the subjects of
mathematical research. The consequence is, that when we have
resolved by a single algebraical computation all the problems of the
same kind proposed, in the utmost generality of which they are sus
ceptible ; the application of the investigation to all particular cases
requires no more than arithmetical operations.
It is usual, though by no means absolutely necessary, to represent
quantities that are known by the commencing letters of the alphabet,
as a, ^, c, dy &c., and those that are unknottn by the concluding letters
w, Xy t/y z. But it is often convenient, especially as it assists the
memory, to represent any quantity which enters an investigation,
whether known or unknown, by its initial letter ; as Bum by «, pro
dtict by py density by rf, velocity by c, time by t; and so of others.
In addition to the signs already explained as being used in arith
metic, the following symbols and modes of expressing certain opera
tions, are employed in algebra.
The product of two or more quantities is expressed either by in
terposing the sign of multiplication, as a x b x c x d; or by inter
posing dotSy which have the same signification, bs a . b . c .d; or, more
simply, by placing the letters merely in juxtaposition, B&abcd. And
as it is immaterial in what order the multiplication is performed, it is
usual to write the letters after each other in the order in which they would
stand in the alphabet, placing any figures which may occur in the product.
THiP. 11.] AND NOTATION. 53
at the commencement. Thus, the continual product of /2 ^, a;, 7 e,
lod S c, may be written either ^bzl eScy or x ^bl eSc^ &c., but
it is preferable and usual to write them i^ b e e x. The figures
% 3, 7 and 42, by which the letters are multiplied, are termed their
mficienti; when a letter is not preceded by any figure (as x in the
example), its coefficient is 1, or unity.
We give the name term to any quantity separated from another by
the sign I or — . When an algebraical expression consists of only
1 term, as 4 a it is called a monomial.
2 „ ac — 4 a & . . . „ binomial*.
8„ o + fc — a c . , ' „ trinomial.
4 „ c + </— SyH X . „ quadrinomial.
{multinomial^
or polynomial .
The signs + and — , which in arithmetic simply indicate the opera
tioos of addition and subtraction, are employed more extensiyely in
algebra, to denote, besides addition and subtraction, any two opera
tioiis or any two states which are as opposed in their nature as addi
tkn tnd subtraction are. And if, in an algebraical process, the
i^ + is prefixed to a quantity to mark that it exists in a certain
itete, pontion, direction, &c., then, whenever the sign — occurs in
eoDoexion with such quantity, it must indicate precisely the con
tiary state, position, &c. and no intermediate one. This is a matter
of pare convention, and not of metaphysical reasoning. Other cha
rKters might have been contrived to denote this opposition ; but they
would be superfluous, because the characters + and — , though ori
gioally restricted to denote addition and subtraction, may safely be
otended to other purposes.
■ignifies any^ ^ theright, \ nifies \ to the left,
(, forwards, J (.backwards.
. . / Increase, "^ — a signi ( Decrease,
^^^ ^ J Money due, f respond j Money owing,
^"^^°*^^ ( Motion upwards, ) mg v Motion doT^nward.
And so on in every species of contrariety. And two such equal quan
tities connected togeUier in any case destroy each other's effect, or are
«8al to nothing, as + a » a'= 0. Thus, if a man has but 10/.
ad at the same time owes 10/. he is worth nothing. And, if a ves
•d which would, otherwise, sail six miles an hour, be carried back
a miles an hoar by a current, it makes no advance.
* When the second term of a binomial has the negative sign ( — ) it is called
54 ADDITION AND SUBTRACTION. [PART T.
Like Bigns are either all poMtive (f ), or all negative ( — ). Aud
unlike are when some are positive and others negative. If there be
no sign before a quantity, the sign + is understood.
Like quantities are such as contain the same letters, and differ only
in their coefficients^ asa — 7 a, or9a6j:+ 2abx^^abx, Un
like quantities are such as contain different letters, as a 6 — c/ + bkx.
When any number of terms are collected under a vinculum or bar,
thus, a f /» — c/^, or inclosed in parentheses thus, (a + 6 — c/)*,
it denotes that the whole quantity so enclosed is to be taken col
lectively, and subjected to whatever operation or process is indicated
by the symbol without the bar or parenthesis ; thus, in the example,
that the compound quantity a f 6 — cfy is to be squared.
An equation is when two sets of quantities which make an equal
aggregate are placed with the sign of equality ( = ) between them ;
As 12 4 5 = 40 — 3, or X f ;/ = a + ft — c d.
The quantities placed on both sides the sign of equality are called
respectively the inemhcrs of the equation.
The sign x indicates that the quantity before which it is placed
is infnite^ or unlimited in its value.
The symbol a , placed between two quantities, signifies that the
second varies as the first : thus, a 7 /> is read a varies as b.
The word therefore being of frequent occurrence in deducing the
successive steps of algebraic processes, is denoted by . • . .
Sect. II. Additimi and Subtraction,
In algebra, the operations answering to those of addition and sub
traction in arithmetic cannot with propriety be called by those names,
as either of them in algebra frequently involves the actual use of both
those processes. They would be better expressed by some general
term denoting the incorporation or striking a balance between the
several quantities employed.
When two simple quantities, or monomialsy are to be added to
gether, as 2 ft and c, or 4 a and 5 a, it is done by connecting them
together with the sign of addition (f); as2ft + c, 4a6a; but
when the quantities are similar, as in the second example, the expres
sion may be simplified by adding together the two coefficients, and
subjoining the common letter, which would then become 9 a.
In like manner, the subtraction of simple quantities, or monomialsy
may be expressed by the sign of subtraction ( — ), or in the case of
simple like quantities by actually subtracting tlie coefficients, and
subjoining the common letter ; thus, the subtraction of ft from a,
/from c, 4 ft c from 7 ft c, is performed thus, a — ft, c — /J 3 ft c. It
not unfrequently occurs that the quantity to be subtracted is greater
ihan the quantity from which it is to be taken, the consideration of
which being totally different from all that the learner has been accus
tomed to in arithmetical operations, involves a difficulty in its appro
CBAP. IL] addition AND SUBTRACTION. 55
bension, bat this will soon be remoTed by an attentive perusal of the
piragraph at page 53, explaining the sense in which the signs
+ and — are used in algebra. Where such occurs, the less quan
tity must be taken from Uie greater, and the negative sign prefixed to
the remamder.
To add together compound quantities, or polynomials, as
(o + 2 61 c) and (4 o f 5 c — rf), write one after the other,
with their proper signs attached, omitting the parentheses, as
fl + 2ftfc + 4a 4 5c— </, which may be simplified by the fur
ther addition or subtraction of its separate terms, and then becomes
5a + 26+ 6 c — </.
Wben quantities to be added are presented promiscuously, it is best
to classify them preyious to their addition.
Thus, 3 aS — 3 * c, + 2 c2, + 4 ^, 3 a^ h 5 6c  2 c^
+ 7aS +56c, f aS — 2cS —46c, 7 a« —86c f 4rf
when arranged become as in the margin, a^ _ 4 6 c + 2 c^
and their sum is readily obtained, as in
thefoorth line. 11 a« — 2 6c + 4 rf
To subtract one compound quantity from another, change the sign
of each term of the subtrahend^ and then proceed as in addition.
Thus, 4 « 6 — 3 6 c And 4a6 — 3c«f 6c
— (2 a 6 — 6 6 c) «. ( a 6 — c^ — 2 6 c)
become 4 a 6 — 3 6c become 4a6 — 3c*+ 6c
— 2a6f66c — tf6f c'^+26c
Resalt 2 a 6 h 3 6 c Result 3a6 — 2c« + d6c
The reason of changing the signs of the subtrahend may be ex
plained as follows : — Let it be required to subtract (c— cQ from (a f 6) ;
let us first subtract c, and the result will be a + 6 — c ; but it is evi
dent that in subtracting c from (a f 6), we have taken away more
than we ought, since it is only c— d which should have been sub
tncted, we have therefore taken away too much by </, and to obtain
the correct result, must add it again to a + 6 — c, which then bc
eomes a + 6 — c + </> in which it is obvious that the signs of the
nbtrahend (c — d) are changed.
In addition and subtraction of algebraic fracti(m8^ the quantities
Buist be reduced to a common denominator, and occasionally undergo
other reductions similar to those in vulgar fractions in arithmetic;
>Qd thee the sum or the difference of the numerators may be placed
om the common denominator, as required.
56 MULTIPLICATION. [PART I.
m, « . c ad be ad + be
..a b e a b a e be
And, J 4.— +— ^1 — 4.4..=
o c ae a b c b a
a2 b^ <r a""' b^^ a« c« b^c"
aoc abc abc abe abc abe
a« + *• + c« + a' ft" 4 a'^ c + ft^ ^^
a ft c
. , a c ad ^ bc^
a — iT ac + c^ arf — dx
c cd c d
ac j ex ^ ad + dx _ a{C'd) f ^ (c + rf)
ed ~ ed
And, *+* **
ft — X ft 4 0?
(ft2 4 aft« + ^«)  (ft«  2fta? 4 ««) 4 ft.
ft  a?2 "■ ii ^0.2'
Sect. III. Multiplication.
1 . To multiply one monomial by another, multiply their coefficients
for the coefficient of the product, and subjoin to it the letters com
posing the two quantities in their natural order, and if the same let
ter occurs in both, add together the two exponents for its exponent
in the product, 1 being understood where no other exponent is ex
pressed. In attaching the proper sign to the product, observe that
the product of two factors having contrary signs is negative^ and of
two factors having the »ame sign positive. Or briefly, that like signs
give 49 and unlike signs — .
Note. — The general rule for the signs may be rendered evident
from the following definition ; multiplication is the finding a magni
tude which has to the multiplicand the proportion of the multiplier
to unity. Hence, the multiplier must be an abstract number, and, if
a simple term, can have neither 4 nor — prefixed to its notation.
Now first, 4 a X 4 »» = 4 »» «> for the quality of a cannot be
altered by increasing or diminishing its value in any proportion;
therefore the product is of the quality pltUy and m a by the definition
is the product of a and m. Secondly, — ax 4m = — mo, for
the same reasons as before, mutatis mutandis. Thirdly, 4 a x — m
has no meaning ; for m must be an abstract number, therefore here
CHIP. II.] MULTIPLICATION. 57
we can baTe no proof. Bat + ax (m ^ n) s= ma ^ na^ n being
Jess than m; for a taken as often as there are units in m is = m a
by the first case ; bat a was to have been taken only as often as there
ire units in m — n; therefore a has been taken too often by the units
in n; consequently a taken n times, or n a, must be subtracted ;
sod of course ma — n a is the true product. Fourthly,
— ax (m — n)a=: — ma f na. For ^ a x m s= — ma (by
case 2); but this, as above, is too great by — na; therefore — ma
with n a subtracted from it is the true product ; but this, by the rule
of subtraction, is = — wi a + w a.
2. To find the product of two pdynomiahy multiply each term of
the one mto all those of the other, following the rule given for mo
uomials.
3. To multiply algebraic fradionSy take the product of the nume
rators for the new numerator, and that of the denominators for the
new denominator.
Examples.
1. iah X bed =i ^ . 5 . ah .cd z= ^0 abed,
2. 8 a« ^» X 4 a* 6 = 8 . 4 . a2 . a* . 6' . * = 32 a« + *
fc' + i = 32o» bK
3. Multiply 2o + *c2*«
By2a — *c + 2ft«
4 a« + 2 a * c — 4 a 5«
— 2a*c— 6«c« +2 5«c
+ Aab^ +2 5»c — 4 6*
Product, 4 a« — *« c« +4^«c4 i^*
U + fc l» H
2 a 6 + ft*
b
o«+ ab a» +2a« b + ab*
ab { b* + a«i^ + 2ad« + ft*
a« + 3a6 + ft« a» •f3a«ft + 3aft« +6»
a + ft
ft
••{:i
« +aft
 a ft  ft«
a« ft«
58 DIVISION. [part I.
a + b a ^b _ ( g j 6) (a — b) _ a" — b\
c rf"" c X d ^ c d
2« "dab 3ac__ IS a be x __ 9 a j?
*o c ^b " tiabe "l
 — —  = 9aa?.
iVb/tf. — From the above examples (4, 5, and 6) we may Icani —
1. Tliat the square of the sum of two quantities is equal to tbe sum
of tbe squares of tbe two quantities together witb twice their pro
duct.
2. That tbe cube of tbe binomial a ^ by is a* f S a' b +
'dal^ {• b\
3. That the product of tbe sum and difference of two quantities is
equal to tbe difference of their squares.
Sect. IV. Division,
1. To divide one monomial by another, divide the coefficient of the
dividend by tbe cocfRcient of tbe divisor for tbe coefficient of the
quotient, and subjoin to it a fraction having for its numerator tbe let
ters composing the dividend, and for its denominator those of the
divisor, and if tbe same letter occurs in both, subtract tbe exponents
for tbe exponent of tbe same letter in tbe quotient ; if the exponents
as well as tbe letters are alike, both the quantities may be struck out.
Tbe same rule applies to tlie signs in division as in multiplication.
2. To divide a polynomial by a monomial^ divide each term of the
polynomial by the monomial according to Rule 1, and connect the
results by their proper signs.
3. To divide two jwlynomials one by tbe other, arrange them witb
respect to tbe powers of tbe same letter, then divide tbe first terms
one by tbe other, and thence will result one term of tbe quotient ;
multiply tbe divisor by this, and subtract the product from tbe divi
dend : proceed witb the remainder in tbe same manner.
4. To divide algebraic /ractiont^ invert the terms of tbe divisor,
and proceed as in multiplication.
Ea!amples,
1. 'ladH'c^Sab^ y^a*'b''c=^id'bc.
U. 15 a ♦ b' ^ 5 a b' = »/ a^'b'' = 3 a b\
8. 12 «^ — 2a f f c r 2 rt = a — c 4 r^ •
2a^
H.r 12j:v l).r//^
i (i X f 12 aw/ — .;? V ^ t '\ .v = h — ^ ^^ =
CHAP. II.] DIVISION. ,59
O
6. Divide af^ ^ Sjp^z ^ Sxsr — sf"* by x ^ z.
x^z)x^^SarZ'\3xz^^z^(a^'^xz'\z^ quotient
— 2ar^2r4 ^x^
xz'^z^
a? jy^ — 2r*
Divide a* — b^ by a — b.
a  6) a» — 6Va* + a^ 6 + a* 6' + a ft* 4 ft* quotient.
a^aH
a* ft an*
a'b''a'b^
Here the second tenn of a^ ft'*
the dividend is brought c^b^ ^ ab^
down to stand over the
corresponding term in a ft* — ft*
the last product. aft* — ft ^
8. Divide 1 by 1 — a?
\^x)\ (1 + 0? + ;r + ar^ + a^ + ^^
1 —a? ^""^
J? — X'
«'
««
*•
• ar^
ar"*
60
INVOLUTION.
[PAKT I.
9.
2d?'
a' H X'
^x
a j X a* ^ x^
U X' a + X _ 2 J? (o + ;p)
X "~ (a* + 0?*) ;p
x'^h'
10.
X' ■\ hx ^* — 6*
F.X
X — b
x'^^hx'^h'' xh {x^hf x{x^h)
;p' + ft ' ft"
a? (;r h ft) (j? — ft) X {x — ft) ^ X '
11. Divide 96 — 6 a* by 6 — 3a. Qiwt. 16 +8a jAa^ ^Hal
12. Divide 10a' + 1 1 a'ft — 19 aftc — 15 a^c ( 3 aft« + ISftc''
— 6 b'c by 3 a ft + 5 a — 5 ft c. Quot, 2 a + ft — 3 c.
13. Divide x" +y" + , by a: f y +  • QtioL a? — y f  .
Sect. V. Involution,
1. To invoice or raise monomials to any proposed power.
Rule, — Involve the coefficient to the power required, for a new
coefficient. Multiply the index of each letter by the index of the
required power. Place each product over its respective letter, and prefix
the coefficient found as above : the result will be the power required.
All the powers of an affirmative quantity will be f : of a ne^ive
quantity, the even powers, as the 2nd, 4th, 6th, &c., will be + ; the
odd powers, as the 3rd, 5th, 7th, &c., will bo — .
To involve fractions, apply these rules to both numerator and
denominator.
The application of these rules is fully exemplified in the following
table of
BOOTS AND POWBBS OP
U0N0MIAL8.
Root
a
b
V*
e
a
a*
I'x
2y
1
a a,"
Square
<^
¥
V*'
a*
a'
1
a'V
*»«'
Cube
a"
ft'
X
«'
a"
8y
1
a"*'
4th Power
a*
V
X/x'
a*
a"
ley
1
a»*'^
5th Power
a"
b'
Vx^
a'
32 y
1
 a*^x"
CHiP. II.] INVOLUTION. 61
^. To involve polyn&miali,
M.'Mnldply the given (quantity into itaelf as many times,
wanting one, as there are units in the index of the required power,
ud the last product will be the power required.
Example,
Cube x±,z and 'Hx — ^z.
X ±,z 2d? — 3j2r
2a? — S^r
x±z
z*'
r±xz
±xz^
QnaTes
3^±^XZ
X ±z
+ Z^ ....
x'±iix'z
± x^z
+ ^
Cobes
\X^±%3?Z
+ Sxz*
■hz\
4 a?' — Qxz
— Qxz \9z'
.4:x'''^l2xz + 9z'*
fix — 3z
Sx^^Uxz^ 18;r;»*
— 12«»;2f + 36a?;»« — 27;r^
The operation required by the preceding rules, however simple in
^ nature, becomes tedious when even a binomial is raised to a
liigii power. In such cases it is usual to employ
Sir Imoc Newton's Rule /or involving a Binomial.
1. Tojind the terms without the coefficients. — The index of the first,
or leading quantity, begins with that of the given power, and de
cwisea continually by 1, in every term to the last; and in the
following quantity the indices of the terms are 0, 1, 2, 3, 4, &c.
2. To find the uncice or coefficients. — The first is always 1, the
•Kond is the index of the power : and, for the others, if the coeffi
cient of any term be multiplied by the index of its leading quantity,
ttd the product be divided by the number of t^rms to that place, it
^giie the coefficient of the term next following*.
ilToiff.— The whole number of terms will bo one more than the
Wex of the given power; and when both terms of the root are + ,
ill the terms of the power will be + ; but if the second term of the
foot he — , all the odd terms will be + , and the CTen terms — .
* This role, expressed in general terms, is as follows : —
(• + 4),  «• +».«"' 6 + n. L=J a«6« + n . !L±i.'LzJa"«i«,&c
2 2 3
The lame theorem applied to fractional exponents, and with a slight modifi
ation, ssrres for the extraction of roots in infinite series ; as will be shown a
^&rtheron.
62 INVOLUTION. [part I.
Eapamples.
1 . Let a '\ X he involved to the fifth power.
The terms, without tho coefRcieDts, will he
a\ a^ Xy d^x\ a^ x\ ax*y x\
and the coefficients will he
^ 5x4 10 X 3 10 X a 5 X 1
' ^' 2 ' 3 ' 4 ' 5 '
and therefore the fifth power is
a' + ba*x + lOa'x h lOa^a?' f 5aa?' f jr\
Here we have, for the sake of perspicuity, exhibited separately
the manner of obtaining the several terms and their respective co
efficients. But in practice the separation of the two operations is
inconvenient. The hest way to ohtain the coefficientB is to perform
the division first, upon either the requisite coefficient or exponent
(one or other of which may always be divided without a remainder),
and to multiply the quotient into the other. Thus, the result may
be obtained at once in a single line, nearly as rapidly as it can be
written down.
2. {x ^ yy ^ x'' \l a^ z ^ 21 X' z' \ 35 a?* z' + 35 x^z^
f 21 x'z^ f 7a?2^ f z*.
3. {x ^ zf :=^ 3^ ^^x^ z \ 28 :i^ ;f*^  56 x^ z' + 70 x' z'
— 50 X' X?^ 4 28 X' ;?«  8 4? ;J^ f ;^^
For Trinomials and Quadrinomiah. — Let two of the terms be
regarded as one^ and the remaining term or terms as the other ; and
proceed as above.
Example.
Involve X \ y — z to the fourth power.
Let X be regarded as one term of the binomial, and y — ^ as the
other : then will U + ^ — zY = {;i? + ( j^ — 2?)}^ = a?* i 4 ar* (y — z)
+ 6 a?*^ (^ — z^ f A: X {y ^ zy + {y — z)\ where the powers of
y — z being expanded by the same rule, and multiplied into their
respective factors, we shall at length have a?* f ^ x^y ^ Ax^ z •\
^x'f — l^a^y z + 6aj»ar' + 4 xf — V2xy'z + ^'Hxyz' —
4 a? ^r* h y* — 4 y** ^ + 6 / z^ — 4 jy 5r* f z\ the fourth power
required.
Had {jX f y) and — z been taken for the two terms of the bino
mial, the result would have been the same.
Note. — The rule for the involution of multinomials is too complex
to be given in this place.
CHiP. II.] KVOLUTION. 8.1
Sbct. VI. Evdutian.
1 . To find the roots of monomials,
iSv/e.— Extract the correspondiDg root of the coefficient for the
Mw coefficient: then multiply the index of the letter or letters hj
the index of the root, (fractionally expressed, as explained at paee 4,)
tk result will he the exponents of the letter or letters to he placed
liter the coefficient for the root required.
Examples.
I Find the fourth root of 81 o*;?*.
First \/ 81 = >/9 = 3, new coefficient.
Then 4 X J = 1, exponent of a; and 8 x J = 2, exponent
of z.
Hence 3 a :2^ is the root required.
2. To find the square root of a polynomial.
Proceed as in the extraction of the square root, in arithmetic, as
explained st page 29.
Examples.
1. Extract the square root of a* + 4 a"* a? + 6 a^ ar^ f 4 a a?* +ar'.
a* + 4 a^x f Qa^x"' f Aax' + x* {a^ + ^ax + ara*
tf« [root required.
4a' a? + 4a=;r^
•io' + aaa?
2 a if
X 2a' x^ + 4a;p»  x
2. Extract the square root of :r* — S a;^ + * 4?* — i a? + ^^ ,
i
X 1
^ 2 ^ 16
a 1
' 2 ^ 16
64 EVOLUTION. [part I.
3. To find the roots of powers in general.
When the power (w) of the root to he extracted is not very
high, the following method may he employed for extracting the nth
root*.
Rule. — Range the several terms in the order of the powers of one
of the unknown quantities ; having done which, extract the nth root
of the first or leading term, which place in the quotient, and cancel
such term ; then bring down the second term for a dividend, which
divide by the term of the root just found, raised to the power of
(n — 1 ), and multiplied by n, the quotient will be the second term of
the root. Then to find the third term of the root, involve the two
already obtained to the nth power as far as the third term, which sub
tract from the third term of the given power for a dividend, which
divided by the same divisor, gives the third term of the required root.
And, in like manner, to find the fourth term of the root, involve those
already obtained to the wth power as far as the fourth term, which
subtracted from the fourth term of the given power, gives a fresh
dividend ; and thus proceed until the root is extracted.
Examples.
1. Extract the cube root of a?« — 6 ar' + 15 a?* — 20 «* + 15 ar*
— 6a? h 1.
^P** 6a?^+ 15a?*20af» + 15d?' — 6« + 1 (a?'— 2^?+ 1
«** [jroat rehired.
3a?*
* 6ar^
irB«6«^*fl2a?* = (a;« — ^f to the third term.
3a?»
j,*_6*» h 16**~20*"+l5*«6*+l=(*>2*+l)*
2. Find the 4th root of 16a* — 96 0=* a? + ^K^a^x — 216aj?'
4 81 x\
16a*— 96a^a?f 216aa?' — 216a;r'» f 81;i?»(2a3^
16 a*
8 a"* X 4= 32 a")* 96 a^^
16 g* — 96a^a? +216o»;r' — 2l6o^81 ** « (2a  3*V
* * * * *
* By this mode of expreision is meant any root whatever; the nile may be
made to apply to any particuhir case by substituting for n, the power of the root
required, which in the first Example above is 3, and in the second is 4.
CHIP. II.] 8UBD8. 05
ilTdtH.— In the higher roots proceed thus : —
For the liqiMdraiey extract the tguare root of the square root.
n nk4 root, ,, cube root of the square root.
„ ei9M root^ „ sq. rt. of the sq. rt. of the sq. rt.
n nmtk rooty ,, cube root o£ the cube root.
Exunples, however, of snch high roots seldom occur in any prac
tical inquirieB.
Sect. VII. Surds.
A Swrdy or irrational quantity, is a quantity under a radical sign or
fractioDal index, the root of which cannot he exactly obtained. (See
Abith. Sect. 9. Evolution.)
Sards, as well as other quantities, may be considered as either
ample or compound, the first being monomials^ as v^S, a*, ^a b\ the
^n poly nomials, as y/3 + v'S, V« + >/*— v'c^* V(«— >/^)'
^a + 3, &c
Radons] quantities may be expressed in theybrm of surds, and the
operation, when effected, often diminishes subsequent labour.
RBDUCTION.
1. To reduce eurds into their simplest expressions.
Ca8B I. If the surd be not fractional, hut consist of integers or in
tegral factors under the radical sign :
Arfe.— Divide the given power by the greatest power, having the
wne mdex, contained therein, that measures it without remainder;
let the quotient be affected by the radical sign, and have the root of
tbe diiieor prefixed as a coefficient, or connected by the sign x .
Examples,
1. v^75 = >/(26 X 3) = v'aS X >/3 = 6 ^S.
a. \/US = X/(U X 7) = V®^ X V = ^ V^.
8. V176 = V(1«X 11) = V16 X V11 = ^V11
4. v^(8af»— 12«*^) = >/4a*(2«8^)=s >/4«* X
v^(2a?— 3^) = 2« v'(2«— 3^).
«. V(56^y + 8^) = VQ^'C^^ f 1) = V^*^ X
Cabb II. ^ M« nircf be fractional, it may be reduced to an equiva
^mteqralone, thus: —
. iMe.— Multiply the numerator of the fraction under the radical
*n bj that power of its denominator whose exponent is one less
F
66 SUBD8. [PABT I.
than the exponent of the surd, and place it as a whole number under
the radical sign, and prefix to it for a coefficient, the coefficient of the
original surd (whether unity or any quantity) divided by the deno
minator of the given fraction taken from under the radical sign.
Note, — This reduction saves the labour of actually dividing by an
approximated root ; and will often enable the student to value any
surd expressions by means of a table of roots of integers.
Ejeamples,
1. v^^; then I x 3 = 3, and  v'S = v'f
a. v^; then 1 x 6 = 5, and V6 = y/^.
V : then 1 x n = n, and  ^n = v^ .
71 n n
VI; then 4 X 5« = 100, and ^ V^^^^ = Vf
3.
3 /2 a
then 2 a X 5a?* = 50 a«*; and — VSOaar*
I 6a!
6. VM = 2 VA» ^^en 2 X 81»= 13122, and
^\V13122 = V18=VM
Case TIL If the denominator o/ the fraction be a binomial or r««f
dualy of which one or both terms are irrational and roots of%quare% :
Rule.^ Multiply this fraction by another which shall have its nume
rator and denominator alike, and each to contain the same two quan
tities as the denominator of the given expression, but connected with
a different sign. ■
Note 1 . — By means of this rule, since any fraction whose nume
rator and denominator are the same, is equal to unity ^ the quantity
to be reduced assumes a new appearance without changing its value ;
while the expression becomes freed from the surds in the denomma
tor, because the product of the sum and difference of two quantities
is equal to the difference of their squares.
Examples.
1 S _ 8 >/5 f >/3 _ 8(v^5 H >/3)
n/5— >/3 >/5— v^3V5 + n/S"" 2
= 4(^/5 + v^3).
2.
3 3 V5 v^2 _3(v^5 y/a)
v^5 f >/2 v^5 + ^^2 V5  ^^2 8
= >/5  >/2.
CHIP. 11.] 9U11M. 67
V10O2V6OIV36 162^^60 ^
= 5I3 ^ = ^=8^^60 = 82^15.
4 >/^^ :^ n/^i^ V5t/3 ^ V5V 3
V5^V3 V^^V3'V5V3 n/5>/3^^
Abte2. — Upon the same general principle any binomial or re
«doal surd, as V^ — V^ ™*y ^® rendered rational by taking
VA;'t v(a— 'B) + V(A"~'BO T V(A""^BO + &c for a
Doltipfier : where the upper signs must be taken with the upper, the
lower with the lower, and the series continued to n terms.
Thus, the expression ^/o' — \/6', multiplied by \jc^ •\' \/ a h'
+ V^+ V**> gives the rational product a* — ft\
^ To reduce Bwrds having different exponenU to equivalent ones
that have a common exponent,
Inrolre the powers reciprocally, according to each other s exponent,
for Dew powers: and let the product of the exponents be the common
exponent
^ofe. — Hence, rational quantities may be reduced to the form of
ttj isaigned root; and roots with rational coefficients may he so
Ktiooed as to be brought entirely under the radical sign.
Examplee.
1. a"and 6", become a"'" or a"" and ft*"'" or A*"".
2. a* and 6% become a * ^ or a* and ft* * * or 5* .
3. si and 2*, become 3* and 2^, or V3' and \/^\ or %/^l
4. (a + ft)*, and (a  ft)*, become ^\/{a + ft)' and »V(«  *)'•
5. The rational quantity a', becomes >/a*, V^> ^/®^ ®' V^"*
6. 4aV5ft, becomes V(^«)' X V^^ V^^o^x V^^ <»•
These and other obvious reductions which will at once suggest
tiiemselTes, being effected, the operations of addition, subtraction, &c.
ire 80 easily performed tipon such surd quantities as usually occur,
^ it will suffice merely to present a few examples without
<J«tailing rules.
f2
68
SURD9.
[PABT I.
ADDITION.
EofA. v'SH v'18=s/(4.2)+v'(9.2) = 2v^2 + 3^/2=6>/2.
2. Add together v^54, ^/^, andv^^.
^^64 = >/(9 . 6) = v'9 X >/6 = 3 v'e ] The sum of these is
^/* =n/A=>/Ax >/6 = *n/6 (3hi + )^/6
^/A = n/(V* • i) = >/ « = J n/6 J = S^lff ^6.
3. >/27a«ar+ v'Sa'ajrs v'COa* . 3«) + (a^ . 3«) =
3a»v'3« + a>/3a? = (3a« + a)N/3aT.
SaVJ + fl" V* = (8« + «')V*•
8UBTRACTI0N.
^a?. 1. 2>/50 ^/18 = 2 ^/(25 . 2)  >/(9. 2) = 2 . 6 >/Q
 3 v^2 = (10  3) V2 = 7 V2.
4. V^S^^'^  V10«* « = V(1^5a* . 2a?)  VC^a' • 2«)
= 6aV2« — 2aV^* = SaV^a?.
5. >/46**«— v'20«^ar»= >/(9**. 6a?)— v'(4a«ar*. 6*) =
(3^«2*a?)>/6a?.
a»  c« / 1
\ac/ .
MULTIPLICATION.
^a?. 1. \/lS X 5 V4 = 6 V(18 . 4) = 6 V(4 .2.9) =
6 »y (8 . 9) = 6 . 2 V^ = 10 V^
2. ^/■^xiVA = f.i^/(i.^Jff) = i^/^^=i^/(^.^) =
i>/:^(fty = iAV36 = ^v'36.
3. a* X a^ = a^"*"i = a'A"*""^ = a+^,
4. (a? + z)^ X (a? f ^F = (« + ;?)*"^* = (« + ^)^.
5. (a? + >/y) X (a? — Vy) = a?» — y.
6. (« + >/y)* X (ar  ^y)i =^{0?^ y)^.
'^. jjr" X 3^ = «" *" = j2r "*" .
CHAf, II.] SURDS. 69
10. (T X (f^ =€r X . = flT— .
a
11. V— ax >/ — asv'aV— 1 X v'a>/— l=ax— 1=— a.
12. n/— ax^/ — ft= v'a>/— 1 X v'5>/— l = >/<lft X — 1
DIVISION.
iSp.l. VlOOO5aV^ = lVT =^V250 = 2V(125.2)
= 10 V2.
=«V(A8) = H.fV8 = MV3.
8. «i ^ a?* = a?^* =: a?"^ = 1 ^ aj*.
II II »— II
4. /^ « = «• = «— .
6. "^^^/(^*«^)^^ ^(a^) = ^^x
INVOLUTION.
&.I. (a*)«=:..a*+* = ia*=:iV^
2 (J^/iy = ^i•i•^/(J•i•i) = :5V^/4 = lV^/(4•)
8. (8 + ^/5)« = {(3 + V6) (3 + >/6)} = 14 + 6 ^/5.
4. (a v'J)* = «'3a» >/* + 3a66v'*.
EVOLUTION.
i&.l. v'lO' = v'lOOO = V(100 . 10) = VIOO X vio
= 10^/10.
8. V31aV« = V(91<**y*5^«) = ^«Wy'^•
3. ^(a* — 4 a >/5 + ^) = a — 3 ^/5, the operation being
P^ormed as in the arithmetical extraction of the square root
Aofe. — The $quare root of a binomial or residual a ± 5, or even
^ i tiinomial or quadrinomial, may oflten be conveniently ex
*neted thus:— Take d = V(a» — ft*); then >/(a ± ft) =
V ^ ± V ^^' ^"^ "^ evident: for, if /y/^^ ±
70 SIlfPLS EQUATIONS. [PABT I.
A / — — be squared, it will give a 4 >/(a* — <^) or a + d, as it
ought : and, in like manner, the square of a / — a / —5 — *
is a — >/(a^ — d^)i or a — 6.
Ex. 1. Find the square root of 3 j 2 v'jJ.
Here a = 3, ft = 2 V^, </ = ^(9 — 8) = 1,
= Vi + N/f = >/2 + n/1 = 1 + >/2.
2. Find the square root of 6 * 2 >/5.
Here a = 6, ft = 2 >/5, rf == >/(86  20) = Vl« = ^
3. Find the square root of 6 + >/8 — >/12 — >/24.
Here a = 6 + ^/8, ft = >/ia f n/24, rf = v^(6 + >/8)» —
(n/12 + v^24)^ = ^/(44 f 12 ^^8 — 36  2 ^^12 . 24) =:
>/(44  36 + 12 n/8  12 s/8) = v^8.
o + rf 6 + 2 ^^8 Q.,o A^ — ^
Conseq. y = ^^^ = 3 + >/8, and ^ =
6 + >/8v/8 .
2 =^
But (Ex. 1), v^(3 f 2 >/2) =^ ^^(3 f ^/8) = I f >/2.
Therefore the root required is 1 + >/2 — >/8*.
Sect. VIII. Simple Equations.
An algebraic equation is an expression by which two quantities,
called members (whether simple or compound), are indicated to be
equal to each other, by means of the sign of equality (=) placed be
tween them.
In equations consisting of known and unknown quantities, when the
unknown quantity is expressed by a simple power, as Xy x\ x\ &c,
they are called simple equations^ generally ; and particularly, ample
or pure quadratics^ cubics^ &c. according to the exponent of the nn
* For the cube and higher rooto of binomials, &c. the reader may oontult the
treatises on Algebra, by Madaurin, Emerson, Lacroix, Bonnyoastle, J. R.
Young, and Bine.
CliP.U.] 8IMPLI EQUATIONS. 71
known quantity. But when the unknown quantity appears in two
or more different powers in the same equation, it is named an ad^
/eetorf equation. Thus a;* = a + 15, is a simple quadratic equation :
2^ f a f = 6, an adfected quadratic.
It is the former class of equations that we shall first consider.
The reduction of an equation consists in so managing its terms,
tint, at the end of the process, the unknown quantity may stand
•lone) and in its first power, on one side of the sign =, with the
ham quantities, whether denoted by letters or figures, on the other.
Thus, what was previously unknown is now affirmed to be equ4d to
the aggregate of the terms in the second number of the equation.
^In general, the unknown quantity is disengaged from the known
wet, hfpefforming upon both members the revbbsb opbbations,"*
to those indicated by the equation, whateyer they may be. Thus,
If any known quantity be found added to the unknown quantity,
jetitbesabtracted from both members or sides of the equation; or
if any sacb Quantity be found subtracted, let it be addedf .
If the nuKnown quantity have a multiplier, let the equation be
dJTided by it.
If it be found divided by any quantity, let that become the mul
tiplier.
If any power of the unknown quantity be given, take the corre
ipondiug root ; or if any root, find the corresponding power.
If the unknown quantity be found in the terms of a proportion
[ir^ Sect. 10), let the respective products of the means and ex
^es constitute an equation ; and then apply the general principle,
ts above.
Examples.
1 Given x — 3 + 5=9, to find x.
Otherwise, in appearance only, not in effect.
By transposing the 3, and changing its sign, d; + 5 = 9 + 3.
By transposng the 5, and changing its sign, d; = 9 + 9 —
*=: 7.
^ Given 3 « + 5 = 20, to find x.
First, by transposing the 5, 3 j; =x 20 — 5 «= 15.
by dividing by 3, j; =s y ss 5.
* This timple directioii, comprehending the seven or eight particular rules
^the rednction oi equations given by most writers on algebra, from the time
*f Ncvtoo down to the present day, is due to Dr. Hutton. It is obviously
^Knded npon the mathematical axiom, that equal operations performed upon
ifi*! things produce equal results.
t These two operations constitute what is usually denominated trantpotUum^
^cenae the operation of thus adding or subtracting any quantity from each side
*f the eqoatiim is moat simply perwnned by tron^poiinf it from one side of the
■fisiion to Che other, and dianging iu sign.
72 SIMPLE EQUATIONS. [PAI
3. Given  f </= 3 6 — 2<:, to findd?.
First, transposing dy  = 36 — 2c + </.
a
Then, multiplying by a, xs=3ab^^ae{ad,
4. Given V(3* + 4) f 2 = 6, to find a.
First, transposing the 2, 51/(^* + ^) = ® ~ ^ = ^•
Then, cubing, 3« + 4 = 4^* = 64.
Then, transposing the 4, 3 « = 64 — 4 = 60.
Lastly, dividing by 3, a? = ^o = 20.
5. Given 4a^— 56 = 3£^;p + 4tf, to find a.
First, transposing 5 b and S da^ 4aa; — 3(f;p = 56 +
Then, by collecting the coefficients, (4 a — Sd)a!^bb ^
.'. by dividing by 4 a — 3 </, a? = r ry
4 a — ott
6. Given a; + j^^ — j^^=3, to find a.
^f^YS ^ ^*^2 = I 80* + 24x  aO« = 360.
4 X 6 X 6, we have ) ^
That is, collecting the coefficients, 34 a; = 360.
.. dividing by 34, a? = ^ = i^ = 10 J^.
7. Given ^x : a : : 6 6 : 3 e, to find x.
Mult means and extremes, ^ c a; s: 5 a 6,
T%. .,. I o ..1 « 20 a 6
Dividing byfc, a;=:5aori<? = ^ •
8. Given a '\ a =^ >/d^ 4 « >/(4 6 + a?'), to find a?.
First, by squaring, we have, a' + 2 a a? + «' = <
a? v^(4 6* + a?^)
Then, striking out a* from both sides, 2 a a? f ^
X >/(4 6» ia:^)
dividing by a?, 2 a + a? = >/ (4 6* 4 a?*)
squaring, 4 a' + 4 a a? + a:** = 4 6* + a?^
striking out a^ and ) ^, _ ,
transposing 4a%)
A' M K . 46^4a« 6'
dividing by 4 a, a: = = a.
4 a a
9. Given \/ca? — ac = 6 H y^* — «> to find a?.
First, dividing by v'^ — «> we have ^/c = —z — — — ^.
CHi?. n.] BXTBBMINATION. 73
V'C— I h
tranqK>8iiigthe 1, ^/c — I, or
1 ^{x  a)
mTerting and transpodDg the fractions, — ^—7 "^sr
moltiplying by d, >/(« — a) =
v'c — 1
sqaanng both sides, 4? — a = — ; — —r
iln^.^:
10. GiTen 18 — >/3 « = >/l3 f 8 ar, to find a?.
An$. X = 12.
8
11. GiTen y + ^/ 4 + y* = 7/4 . ^8) > ^ ^°^ y*
12. GWen J (* + 1) + i (* + ^) == H« + ^) + 10> ^ fi°<l*
il9M. d? = 41.
13. Giyen^^A/^^ s/(a:— 1) :; 3 : 1, to find a:. AnB,\\.
14 Given (6* + «*)» = (a» + ^)*, to find «.
EZTBBMINATION.
Wlen two or more unknown quantities occur in the consideration
of ID algebraical problem, they are determinable by a series of given
ixlepeDdent equations. In order, however, that specific and finite
Miotioos may be obtained, this condition must be observed, that there
^gwm as many independent equations as there are unknown quan
^. For, if tibe number of independent equations be fewer than
tlie unknown quantities, the question proposed will be susceptible of
ID indefinite number of solutions'!' : while, on the other hand, a
greater number of independent equations than of unknown quan
tities, indicates the impossibility or the absurdity of the thing at
ten^.
Where two unknown quantities are to be determined from two in
^^dent equations, one or other of the following rules may be em
^vaknown quantities by two equations ; and so on.
Thus, if J? + %/y = « i >/b
and a — ^v = c — ^d
Then #«a,y»6, sr = o, «=«rf.
74 EXTBRMINATION. [PABT I,
Bide 1 . — Find the value of oDe of the uDknown letters in each of
the given equations ; make those two values equal to one another in
a third equation, and from thence deduce the value of the other un
known letter. This substituted for it in either of the former equa
tions, will lead to the determination of the first unknown quantity.
2. Find the value of either of the unknown quantities in one of
the equations, and substitute this value for it in the other equation :
80 will the other unknown quantity become known, and then the first,
as before.
3. Or, after due reduction when requisite, multiply the first equa
tion by the coefficient of one of the unknown quantities in the second
equation, and the second equation by the coefficient of the same un
known quantity in the first equation : then the addition or subtrac
tion of the resulting equations (according as the signs of the unknown
quantity whose coefficients are now made equal, are unlike or like)
will exterminate that unknown quantity, and lead to the determina
tion of the other by former rules.
Notes. — The third rule is usually the most commodious and expe
ditious in practice.
The same precepts may be applied, mutatis mutandis^ to equations
comprising three, four, or more unknown quantities : and they often
serve to depress equations, or reduce them from a higher to a lower
degree.
Examples.
1. Given d^jf* h 3y = 41, and Sar* — 4y = 12, to find x and y.
1st equa. x by 3, gives lHaP f 9y = 123
2ndequa. x by 4, gives 12 j?* — 16 y = 48.
The difference of these, 25 y = 75, whence y = 3.
Then, from 2nd equa., 3 jr» = 12 + 4y = 12 j 12 = 24
Whence dividing by 3, ar* = 8, or a? =: 2.
Ex. 2. Given a? h y r xr = 53, ;i? f 2y + 3;8f = 105, and
« + 3y + 4;2r= 134.
1. X { y h J2r= 53
2. 0? 4 2y 4 34r=: 106
3. 0? f 3y + 4;t= 184
4. 1st equa. taken from 2nd, gives y f 2« = 52
5. 2nd equa. taken from 8rd, y h jp = 29
6. 5th equa. taken from 4th, jt ss 28
7. 6th equa. taken from 5th, y = 6
8. 5th equa. taken from 1st, x = 24.
Ex, 3. Given x^yssa, x^ zssb^ y \~z^Cy to find Xy y, and s.
1. ar fy = a
2. X ^z= b
3. 1/ ^ z^si c
(IIP. II.] OSNUUL PROBLEMS. 75
4. lft + 2nd + 3rd, gives 2ar + 2y + '^^^==0+ i + c.
5. Ha]f 4th equa. gives ^ry + ^ = ia + 4^ + i^
6. 3rd equa. taken from 5th, gives ^=af5— Ic.
7. dud eqna. taken from 6tfa, ^ = 1^ — 1^ + 1 c
8. 1st eqoa. taken from 6tfa, Jtrss _a + i^ \ ic.
Ex, 4. Given ax \by^ e^ and o^^p f ^'y as ^, to find 47 and y.
cV — h<f , atf — ca^
&.6. Given ajp f ^y + ^Jjzrassrf, c^x + V y rf^ z ^ df of' x +
5"^ + c^'ar =: rf" to find a?, y, and ;2r.
^""a^V  ac'6" f CO' ft''  6aV' + hifa^'  c5V
'"aftV' — ac'ft" +ca'ft"fta'c" + ftc'a"cA'a'''
& 6. Given d:(;p h y + ^) = 18]
y (^ + y r ^) = 27 I to find a?, y, and z.
^(^f^ + 4 = 86 J
iln«. :i?=:2, y = 8, j2raB4.
^.7. Given (a? + y) « 60, and(ay+y) = 2^, to finda?andy.
y ^
Ans. ^ = 10, y = 2.
^. 8. Given 4* + 4y + i;? = 62]
l^+yHi^ = 47[ to find a?, y, and 4?.
ilw*. a? = 24, y = 60, ;2r = 120.
SOLUTION OP GENERAL PROBLEKS.
A general algebraic problem is that in which all the qaantities
^^Bconed, both known and unknown, are expressed by letters, or
^general characters. Not only such problems as have their
ttnditions proposed in general terms are here implied ; but every
^^f^cdar numeral problem may be made general^ by substituting
^'^ for the known quantities concerned in it ; when this is done,
^ problem which was originally proposed in a particular form
^^^^taes general.
In solving a problem algebraically, some letter of the alphabet must
w mbstituted for an unknown quantity. And if there be more
QBbown quantities than one, the second, third, &c., must either be
^^piwied by means of their dependence upon the first and one or
<W9 of the data conjointly, or by so many distinct letters. Thus,
^>Hay separate equations will be obtained, the resolution of which.
76 OBNBRAL PB0BLBM8. [PABT I,
by some of the foregoing rules, will lead to the determination of the
quantities required.
ExampU$.
1. Given the sum of two magnitudes, and the difference of their
squares, to find those magnitudes separately.
Let the given sum be denoted by «, the difference of the squares
by d; and let the two magnitudes be represented by x and y
respectively.
Then, Uie first condition of the problem expressed algebraically
is « + y = *.
And the second is oj' — ^ = d.
Equa. 2 divided by equa. I, gives x^y=i
Equa. 1 added to equa. 8, gives 2 a; =  + « =
Equa. 4 divided by 2, gives x =
Equa. 5 taken from equa. 1, gives y^zs^
9 8
%8
«* + D «•— D
^8 ^8
To apply this general solution to a particular example, suppose the
sum to be 6, and the difference of the squares 12. Then » = 6
and D= 12,
«« + D 86 + 12 48 ,
and X = —  — = —  — = — = 4.
2« 12 12
«* — D 8612 24 ^
and y = — r^ — = — rr — = — = 2.
^ 2« 12 12
Suppose, again, 8 = 6, d = 5 :
. 25+6 ^ . 266 ^
then X = — Jq— = 8, and y = — Jq" "^ ^
Ex. 2. Given the product of two numbers, and their quotient, to
find the numbers.
Let the given product be represented by/?, the quotient by ^; and
the required numbers by x and y, as before.
Then we have, 1. xy^=^p^
X
and 2.  = ^.
V
Equa. 2 X by y, gives, xssqy
Substituting this value) ^^ ^^
of X for it in equa. 1 J ^^ ^
P
Dividing by ^, y* = —
Extracting the square root, y = a/^
CHI?. 11.] QUADRATICS. 77
Then, \j snbstitation, a? = yy = ^ ^ /?. = ^ /?^ = ^//^y•
Suppose the prodnct were 50 and the qaotient 2.
Theny=^^=^^= V26 = 5, and «= V/?y =
^/lOO = 10.
Agun, rappoee the product 36, and the qaotient 2 .
Then y = y^ = ^^ =^/16 = 4,anda^=V;>y=:
V81 = 9.
Ex. 3. OiTen the sum (<) of two numbers, and the sum of their
aqoues s, to find those numbers.
iw. « = J * + J y^ 2 8 — ««, and^ = J « — J v^2 s — **.
£x. 4. The sum and product of two numbers are equal, and if to
eitber snm or product the sum of the squares be added, the result
inll be 13. What are the numbers l—Ans. each = 2.
Ex.5, The square of the greater of two numbers multiplied into
& leaB, produces 75 ; and the square of the less multiplied into the
prater produces 45. What are the numbers ?
Bx, 6. A man has six sons whose successive ages differ by four
Jem, and the eldest is thrice as old as the youngest. Required their
•emalages?— iliw. 10, 14, 18, 23, 26, and 80 years.
Sect. IX. Quadratic Equaiionx.
QuiDRATic Equations are such as contain the square of the unknown
^UDtitj, and which, after due redaction, may be made to assume the
Seseral form Adr' + B« + <' = 0; then dividing by a, the coefficient
^the first term, there results «^   « H  = 0, or, making pss ^
A A
9 = , we have
a!» + jD« + y = (1)
*n equation which may represent all those of the second degree,
J^Vidg being known numbers, either positive or ne^tive.
Ut a be a number or quantity which, when substituted for ^, rcn
fe8^+/?a?f ^ = 0; thenar ^ pa f ^ = 0, or y== ^a^^pa.
^Wqaently j?' + jt> « + y, is the same thing as;i^ — a*+J^^~"
^«, or as (« + a) (« — a) ^ p (a ■— a), or, lastly, as (a? — a)
(«+a + ;?).
The inqairy, then, is reduced to this, viz. to find all the values of
* which shall render the product of the above two factors equal to
*^g. This will evidently be the case when either of the factors
78 QUADRATIdS. [?ART I.
is = ; but in no other case. Hence, we have x^Oy ^0, and
« h a f J^ = 0, or ;i? s=s a, and a? s= — a — p*.
And hence we may conclude —
1. That every equation of the second degree whose conditions are
satisfied by one value (a) of a?, admits also of another value ( — a — p).
These values are called the roots of the quadratic equation.
2. The sum of the two roots a and — a — p is =^ — p; their pro
duct is — a^ — apy which as appears above is = g. So that the eoefi
cienty jD, of the second term is the sum of the roots with a contrary
sign ; and the known term^ q^ is their product,
3. It is easy to constitute a quadratic equation whose roots shall
be any given quantities b and d. It is evidently a^ — (6  rf)
X khdzzz 0.
4. The determination of the roots of the proposed equation (1) is
equivalent to the finding two numbers whose sum is — jp, and pro
duct q.
5. If the roots b and d are equal, then the factors x — b and x^d
are equal ; and ar •\' p x ^ q \s the square of one of them.
To solve a quadratic equation of the form a? •\' px ^ ^ =: 0, let it
be considered that the square of ^ +  j9 is a trinomial, a^ + /> a? + \jfy
of which the first two terms agree with the first two terms of the
given equation, or with the first member of that equation when q is
transposed.
That is, with aj^ + jt? a? = — y.
Let then \f^he added, we have
0^ { px ^^p^ ^ \p^ — q
of which the first member is a complete square.
Its root is a? f ^jt? = ± >/ {\p^ — q)
and consequently iP = — i j» ± 'J{\j^ — q)
otherwise, from number 2 above, we have
X \ of := — p and xaf =^q.
Taking 4 times the second of these equations from the square of
the first, there remains a^ ^^xaf + ^=jp* — 4j'
Whence, by taking the root, a? — a?' = >/(/?* — 4 y)
Half this added to half equa. 1, gives
And the same taken from half equa. 1, gives
which two values of a evidently agree with the preceding.
It would be easy to analyze the several cases which may arise, ac^
cording to the different signs and different values, of/? and q. Bat
* If it be affirmed that the given equation admitN of another valae of jt^
besides the above, h for instance, it may be proved as before that # — 6 mnst
be of the number of the factors of «■ + p * + 9, or of (* — a) (* + fl + ji).
But » — a and jr f a f p being prime to each other, or having no oommon
factor, their product cannot have any other factor than they. Conieiiuently 6
must either be equal toaorto — a— p; and the number of roots is restricted
to two.
CHIP. II.]
QUADRATICS.
79
tliese need not here
be
traced. It is evident that whether there he
flTCB
1.
2.
3.
4.
^ + />* = q
3^ ^ px == q
3^ + px = —
3^  px = —
9
The general method of solution is by completing the square^ that
ii, adding the square of />, to both members of the equation, and
tlien extracting the root.
It may fariber be obsenred that all equations in which there are
two tenns inTolving the unknown quantity or any ftinction of it, and
tbe index of one double that of the other, may be solved as qua
dntics, by completing the square. Thus :
It M
^ + pi» = ^, «*■ i />«* = y, a?^ ± /> a?^ = y, {jf^ ^ p X \ qf
±(^ {px ^ q) =zr,{af^ ^ ary ± (aj*" — j^) = y, &c., are
of tbe same form as quadratics, and admit of a like determination of
die unknown quantity. Many equations, also, in which more than
one unknown quantity are invoWed, may be reduced to lower dimen
Boni by completing the square and reducing ; such, for example, as
(!? + /)» ± /» (:2r^ + y) = ^, 3 ± ^ = ^, and so on.
Sr y
N(fte, — In some cases a quadratic equation may be conveniently
nhed without dividing by the coefficient of the square, and thus
vithoQt introducing fractions. To solve the general equation a o^ ±
*«=<;, for example, multiply the whole by 4 a, whence 4 a* a?® di
iahx = 4 a <;, adding ^ to complete the square, 4ta^aP ± iab x
+ ft' = 4 a'c f P taking the square root, 2 a « ± J = di
*J(iae + ^); whence x = ^ — ^ }" : which will
Krre for a genera] theorem.
Examples,
I Giren x^ — 8 x + 10 = 19, to find x,
transnosing the 10, dj» — 8a? = 19 — 10 = 9
completing the square, a?^ — 8« + 16s= 9fl6 = 26
extracting the root, a? — 4 = ± 6
consequently a? = 4±:5 = 9or — 1.
^ Given ^ — ^ — , to find the values of x,
X ST 9
multiplying by «*, 10 a? — 14 + 2 a? = — — ,
tnnsposing, ^ «* — 12a? = — 14,
dividing by V, ««^«=^,
cmplet. squ. x»  44 oi + (^^ = iH  « = i¥r>
extract, root, « — 4i » ± fr,
tmnspottng, x^^±^^S or ^.
80 QUADRATICS. [PARI I.
3. Given «*f2a? + 4 >/«* h 2 a? 4 1 = 44, to find ».
adding 1, we have (j;*  2a: + 1) + 4 ^{x^ + 24? 4 1) =45
complet squ. (a:* f 2a! + 1) 4 4 >/(jj* + 2a? f + ^ = 49
extract, root, ^{^ 2a?Hl)H2 = ±7
transposing the 2, ^{o^ f 2a? +1) = ±7 — 2 = 5 or — 9
that is,a?4l = 5 or — 9
hence a? = 4 or — 10.
4. Given a?" — 2 aa?^ = c, to find x.
n
complet. squ. a;" — 2aa?^ a?=:cfo^
extract, root, a:^ — a = ±: >/ (c + a*)
n
transposing, a?^ = a di ^^^ (c + o^)
consequently, x =^ {a '±. y/c f a*)".
•E^ 4;r
6. Given ^4 — = 12, and a? — y = 2, to find a? and y.
tr y
3^ X
complet. squ. in equa. 1, — f444=16.
Extracting root  f 2 = ±: 4 : whence  = 2 or — 6, and
a?= 2y or — 6y.
Suhstituting the former value of x in the 2nd equa., it becomes
2^ — y = 2, or y = 2 ; whence a? = 4.
Again, substituting the 2nd value of a;, in equa. 2, it becomes
— 6y — y or — 7y = 2; whence y = — ^, and a? = + V .
6. Given a?*y' — 5 = 4 a? ^, and i a? y = ^ y*, to find a? and y,
equa. 1, by transposition, becomes a?*y* — 4a?y = 6
completing the square, a^}^ — 4a?y + 4 = 9
extracting the root, a?y — 2 == ± 3
whence a?y = 5 or — 1.
Substituting the first of these values for a?j^ in equa. 2, it becomes
. y = ^ : whence ^ = 1 and a? = 5.
Substituting the 2nd value in the same equation, it becomes
I y = — J : whence y = — \j\ = — i V^^> *°^ a? = — 1
ilfw. a? = ± 3, or ± >/^.
8. A man travelled 105 miles at a uniform rate, and then found
that if he had not travelled so fast by two miles an hour, he would
have been six hours longer in performing the same journey. How
many miles did he travel per hour ?
Am. 7 miles per hour.
CHAP. II.] EQUi^TTONS. 81
9. Find two socb numbers ttmt the sum, product, and difference
of their squares may be equal.
Ans. \ + i >/5, and ^ + 5 >/5.
10. A waterman wbo can row eleven miles an hour with tlie tide,
ud two miles an hour against it, rows five miles up a river and back
sgaJD in three hours : now, supposing the tide to run uniformly the
same way during these tbree hoars, it is required to find its velocity ?
Ans. 4J^ miles per hour.
Sect. X. Equations in General.
Equahons in general may be prepared or constituted by the
maltipiication of factors, as we have shown in quadratics. Thus,
snppose the values of the unknown quantity x in any equation were
to be expressed by a, b, c, </, &c., that is, let a; = a, a? = 6, ^ = c,
p=(i, &c., disjunctively, then will x — a=:0, ar — ft = 0, ar — <?=0,
*rf=0, &c., be the simple radical equations of which those of
tiie higher orders are composed. Then, as the product of any two
of th^ gives a gvadratic equation, so the product of any three of
them, as (4? — a) (4J — b) {x — c) =: 0, will give a cubic equation,
or one of three dimensions. And the product of four of them will
coosdtute a biquadratic equation^ or one of four dimensions ; and so
OD. Therefore, in general, the highest dimension of the unknown
fiOHHtff X is equal to the number of simple equations that are mul
fiplied together to produce it.
When any equation equivalent to this biquadratic (x — a) (jx — b)
(* t) {x — d) = is proposed to be resolved, the whole difficulty
wniists in finding the simple equations a? — a = 0, x — 6 = 0,
Jfc=0, X — ^ = 0, by whose multiplication it is produced; for
each of these simple equations gives one of the values of x^ and one
wlotion of the proposed equation. For, if any of the values of x
<ledoccd from those simple equations be substituted in the proposed
equation, in place of x^ then all the terms of that equation will
^iih, and the whole be found equal to nothing. Because when it
i« sopposed that ^ =: a, or x^b^ or ;p = c, or x z= dy then the
product (4? — a) (x — b) (a? — c) (a? — d) vanishes, because one
of the factors is equal to nothing. There arc therefore four supposi
tions that give (x — a) {x — b) {x — c) {x — <^) = 0, according
to the proposed equation ; that is, there are four roots of the pro
posed equation. And after the same manner any other eauation
•dmita of as many solutions as there are simple equations multiplied
^ one another that produce it, or as many as there are units in the
behest exponent of the unknown quantity in the proposed equation.
Bnt as there are no other quantities whatsoever besides these four
K ^ c, d^) that, substituted in the proposed product in the place of
^ wfll make that product vanish ; therefore, the equation {x — a)
(*h){x — #r) (;p — df) = 0, cannot possibly have more than these
82 EQUATIONS. [part
four roots, and cannot admit of more solutions than four. If ^
substitute in that product a quantity neither equal to a, nor 6, nor
nor dy which suppose e, then since neither e — a, c— i, e — c, n
e — dyis equal to nothing; their product cannot be equal to nothir
but must be some real product : and, therefore, there is no suppo
tion beside one of the aforesaid four, that gives a just value of
according to the proposed equation. So that it can have no mc
than these four roots. And after the same manner it appears, i\
no equation can have more roots than it contains dimensions of i
unknown quantity.
To make all this still plainer by an example, in numbers, suppc
the equation to be resolved to be x^ — 10 ar* + 35 ;ir — 50 j? + 24 =
and that we discover that this equation is the same with the prodi
of (a? — 1) (j?— Q) {x — 3) {x— 4), then we certainly infer tl
the four values of a: are 1, 2, 3, 4 ; seeing any of these numbe
placed for Xy makes that product, and consequently a?' — 10 dr*
35 x^ — 50ii? + 24, equal to nothing, according to the propos
equation. And it is certain that there can be no other values of
besides these four : for when we substitute any other number for
in those factors ii? — 1, d? — 2, x — 3, ;r — 4, none of them vanL
and therefore their product cannot be equal to nothing, accoi'ding
the equation.
A vai'iety of rules, some of them very ingenious, for the soluti
of equations, may be found in the best writers on Algebra*; 1
we shall simply exhibit the easy rule of TrialandError, as it is giv
by Dr. Hutton^ in the 1st vol. of his ** Course of Mathematics."
Rule for the general solution of Equations hy TrialandError,
" 1 . Find, by trial, two numbers, as near the true root as possib
and substitute them in the given equation instead of the unknot
quantity ; marking the errors which arise from each of them.
" 2. Multiply the difference of the two numbers, found by tri
by the least error, and divide the product by the difference of t
errors, when they are alike, but by their sum when they are nnlil
Or say, as the difference or sum of the errors is to the differei
of the two numbers, so is the least error to the* correction of
supposed number.
'^ 3. Add the quotient, last found, to the number belonging to t
least error, when that number is too little, but subtract it when t
great, and the result will give the true root nearly,
" 4. Take this root and the nearest of the two former, or a
other that may be found nearer ; and, by proceeding in like mann
a root will be had still nearer than before ; and so on to any degi
of exactness required.
" Note — It is best to employ always two assumed numbers tl
shall differ from each other only by unity in the last figure on I
right; because then, the difference, or multiplier, is 1."
* i'^ee the trentiseii of I/acroix, BonnycMtle, Wriod, J. R. Vming, &c.
CBIMI.]
BQUATIONS.
83
Example,
To find the root of the eohic equation ^r* + ^v' + x = 1 00, or the
kloe of X in it.
Here it is soon found that x
L^ between 4 and 5. Assume,
b^refore, these two numbers,
M^ the operation will be as fol
iSup.
le
64
84
16
2nd Sup.
X
. sums
. errors
5
25
125
155
^55
the som of which is 71.
TbcnasTl : 1 :: 16 : 225.
Hence x = 4*225 nearly.
Again, suppose 4*2 and 4*3,
and repeat the work as fol
lows :
\9tSup.
42
1764
74088
95928
—4072
X
x'
x'
2nd Sup.
43
1849
79507
sums . . 102297
errors . . +2* 297
the sum of which is 6*369.
As 6369 : 1 :: 2297 : 0036
This taken from . . 4*300
leaves x nearly = 4*264
Agiio, soppose 4264 and 4*265, and work as follows :
4264
18'181696
7:'526752
W972448
0027552
X
x^
x"
sums
4265
18190225
77581310
100036535
+ 0036535
the sum of which is •064087.
Then as 064087 : 001 : : 027552 : 00004299
To this adding ... 4*264
gives X very nearly = 42644299
When one of the roots of an equation has been thus found, then
^t for a dividend the given equation with the known term trans
ited to the unknown side, so as to make the equation equal to
•olkiog ; and for a divisor take x minus the root just determined :
(he quotient will be equal to nothing, and will be a new equation
o 2
84 PR0OBE8SION. [PABT
depressed a degree lower than the former. From this a new Tat
of X may be found : and so on, till the equation is reduced to a q%:
dratic, of which the roots may be found by the proper rules.
Example,
Given the biauadroHc equation, «* — 1 1 «^ f 28 «^ f 36 ^e
144 = 0, to find the four roots.
First, by the above method of TrialandError, we find one oF
roots to be 6 ; then,
a!.6)a?«— lla?'f 28a?^ + 36a; — U4(a?^  5 «'  2*4
a?«— 6 a?'
—
6a?'
f28a?'
+ 30a?^
+
.2 a?''
2 a?*
36 a?
12 a?
24 a? — 144
24 a?— 144
The quotient of which, ar' — 6 a?*' — 2 a? f 24 = 0, is a eMc ^^
tion of which one of the roots is found to be 4 ; then,
r  4 )ar»  6 a?* — 2 a? + 24 (a?^ — a? —
ar»4a?'*
a?' — 2a?
a?' + 4a?
— 6a? + 24
— 6a? + 24
The quotient of which, a?^ — a? — 6 = 0, is a quadratic equation,
which the two roots are readily found to be 3, and — 2 ; thus t J
original biquadratic equation is composed of the factors (a? —
(a?  4), (a?  3), and (a? + 2). Or,
(a? + 2).(a?3).(a?4).(a?6) = a?»l]a?'»+28a^+36a?~144=t
Sect. XI. Progression.
When a series of terms proceed according to an assignable orde
either from less to greater or from greater to less, by continual eqn
<*
CSIP. II.] ARITHMETICAL PB0GRBS8I0N. So
diflerences or by saccessive equal products or quotients, they are said
to fonn a/iro^retttois.
If tbe quantities proceed by successive equidifferences they are
«id to be in Arithmetical Progression, But if they proceed in the
«iDe continued proportion, or by equal multiplications or divisions,
ibev are said to be in Geometrical Progression.
If the terms of a progression successively increase, it is called an
^Kotiing progression : if they successively decrease, it is called a
^ktcmding progression. Thus,
1, 3, 5, 7, 9, &c. form an ascending arithmetical * "^
^2^ 20, 18, 1 6, &c. form a descending arithmetical
> Progrcssi
iion.
1, 3, 9, 27, 81, &c. form an ascending geometrical
^ ^ 1) !» 49 ^^' ^'^1'°^ A descending geometrical /
ABITHMBTICAL PBOOBESSION.
1 Let a be the first term of an arithmetical progression,
d the common difference of the terms,
z the last term,
n the number of terms,
9 the sum of all the terms.
Tbcnei,af(i; a^'Stdj a^^d, &c., is an ascending progression,
and a, a^dj a — Sd!, a^^dj &c., a descending progression.
Hence, in an ascending progression, a 4 (« — 1) <^ ^^ the last term ;
m a descending progression, a — (n — 1 ) </, is the last term.
^Lcta series be a+{a + d) f (a +2 </) + (« + 3 rf).
TlKwne inverted (a + 3rf) f (« + 2</) + (« + <if) + «
^«amofthetwo(2ai3rf)+(2af3<ir)4(2a+8<Qf(2a+8rf)=2*.
Thitii, (2a + 3</) x 4, in this case (a f a + 3rf)n = 2#.
. Cowe^ntly, »=J(« + a"+"37)n, or = i(<» + ^)»»> "°<^ *
>kie = a f 3</. The same would be obtained, if the progression
*^ descending; and let the number of terms be what it may.
3. From the equations ;? = a f (« — 1) <^ * = 4 »* (<• + ^)» ^^^
'=*}«{a ^0 + (n !)<'}» ^6 ™*y readily deduce the following
!^^<Kemi applicable to ascending series. When the series is descend
^ other U)e signs of the terms affected with d must be changed,
I g I *<BiQ8t be taken for z; and vice versd.
« 2«
/9\j * — « «* — fl^ ^zn — ^s 2# — 2na
»— 1 2« — ;» — a n» — n n» — »
(3.)* = a + nrfd=* +i*rf Jrf =
86 OKOMETRICAL PROGRESSION. [pART I.
(4.) «=w(a + ;2r) = (af ^«(/rf)n=:(z^«</ + ^d) n
" Ud
(5.) w = = ; I.
^ ^ a ^ z d
Examples.
1. Required the sum of 20 terms of the progression 1, 3, 5,
7, 9, &c.
Here a =: 1 , </ = *^, w = 20 ; which heing suhstitutcd in the
theorem « = (a + ^7«(/ —  </) 7«, it hecomes « = (1 f 20— 1) '20
= 20 X Hi) = 400, the sum required.
Note, — In any other case the sum of a series of odd numbers
beginning with unity, would be = w^, the square of the number
of terms.
2. The first term of an arithmetical progression is 5, the last
term 41, the sum 299 Required the number of the terms, and the
common difference.
^« 59H ,^ , , ^
Here n = = —  =: 13, the number of tenns,
a I 5r 46
and d = = —  — = 3, the common difference.
w — 1 12
3. There are 8 equidifferent numbers : the least is 4, the greatest
82. What are the numbers ?
TT * z ^ a 32 — 4 ^ , ,. _
Here d = 7 = — z — = 4, the common difference.
n — 1 i
Whence 4, 8, 12, 16, 20, 24, 28, 32, are the numbers.
4. The first term of an arithmetical progression is 3, the number
of terms 50, the sum of the progression 2600. Required the last
term and the common difference.
Here ;2r = — — o = — ; 3 = 104 — 3 = 101, the last term,
n 50
, ^ ;2r — a 101 — 3 ^ .
and d = = — = 2, the common difference.
w — 1 49
5. The sum of six numbers in arithmetical progression is 48 ; and
if the common difference d be multiplied into the less extreme, the
product equals the number of terms. — Required those terms.
Ans. 3, 5, 7, 9, 11, and 13.
GEOMETRICAL PROGRESSION.
Let a be the first term of a geometrical series ;
r the common ratio ;
z the last term ;
n the number of terms ;
s the sum of all the terms.
CHAP. II.]
GEOMETRICAL PROGRESSION.
«:
. am:
tea
Then a, ra» r'o, r* a, r"""'a, is a geometrical progression,
whicli will be tucending or descending^ according as r is an integer
or zfraaioH,
Let the prog, a f ra f r*a + r^a f r*a = «, be x by r, it
becomes ra ^ f^a { r^ a { r^a •\ r^a ^:^rs.
The diff. of these is, — a ^^ r^a^^ rs — ».
But r^a is the last term of the original progression multiplied
by r, or in general terms r"""' a x r^ that is r" a. Consequently
r*<i — a = r* — ».
«T, r"a — a r" — I , ^ ,
whence « = = —a ^ the sum of the series.
r — 1 r — 1
A amikr method will lead to a like expression for «, whatever be
the nlue of fi. If r be a fraction, the expression becomes trans
lormed to # = a.
1 — r
Now from these values of z and s the following theorems may be
deduced.
; * f r;2r — r*.
r r" — 1
w*=
j(r» — 1) _ q(l — r") _ r^j2_l— ^^^ ^^
1 r
r— 1
r" — r
(4.)r
(5.) ji = log y 4 log. ^ log «
log. r
And, if the logarithm of  = N, that of = M, and
tbtofrsR: then
Also, if when r is a fraction, n is infinite, then is r" = 0, and the
^ipKttion for • becomes
(7.) 9 ss , which expression is often of use in the summa
1 — r
tion of infinite series.
h The least of ten terms in geometrical progression is 1, the
'^io 2. Required the greatest term, and the sum.
88 FRACTIONAL AND NEGATIVE EXPONENTS. [PART 1.
Here z = ai^~^ = 1 x ii'* = 512, the greatest term ;
rz —a 2 X 512 — 1
and « =: = = 1023, the sum.
r — 1 1
2. Find the sum of 12 terms of the progression 1, ^, ^, j^*^, &c.
Hei^ , = ^_, = _^pL =^^^^, the sum.
3. Find the sum of the series 1, , ^, , &c., carried to infinity.
Here by Theor. (7.) % = , becomes * = j = ^> ^^*®
sum required.
4. Find the vulgar fraction equivalent to the circulating decimal
•36363636.
This decimal, expressed in the form of a series, is, ^^^^ f T;^Jrt,)
+ TOoVoort + &c., where a = ^^^ , and r = ^J^ .
Consequently, a = = f^^ r tVo = i^' '^*® fraction sought.
5. Find the sum of the descending infinite series 1 — x r jr
 J7* f X\ &C.
Here a = 1, r = — ;u, and s = = ; , the sum req.
And, by way of proof, it will be found that if 1 be divided by
I + ^, the quotient will be the above scries.
6. Of four numbers in geometrical progression tbc product of the
two least is 8, and of the two greatest 128. What are the numbers?
AnB. 2, 4, 8« and 16.
Sect. XII. Fractional and Negative Exponents.
In the preceding sections on powers and surds, we have only
spoken of such powers as have positive and integral quantities for
their exponents, and which are termed direct powers. And we ex
plained that the exponent of the quantity, expressed the number of
times that that quantity was multiplied by itself; so that to denote
the division of any power by its root, we have only to subtract
a"
1 from tbe index or exponent of that power ; thus a" * = — ; and
a"
by continuing to divide by a, we have a'~'"* = a**"* = — ^, and
a* a*
a""^"' = a""' = — ; or generally, a"~* =— ^; that is, to express
the division of one power of a qiiantitt/ by any other potter of the
same quantity ^ we subtract the exponent of the divisor from that of the
dividend.
I
eUiP. II.] FRACTIONAL AND N£OATIVK BXP0NBNT8. 89
If ft b greater than m, then will the quotient ^ = a'' have for
i K^ exponent {p) a positive numher, and is then called a direct power
^.*ta'y bat if m is greater than n, p will he negaUive^ and a"' is then
a*
^r^led an inverse power of ay thus j = a*""* = «*, is a direct
a*
^>«wer, but — j = a'~* = a~^ is an inverse power.
Now, to arrive at a just idea of the value of a quantity with a
x^^gadye exponent or of an inverse power, let us successively subtract
mmwityfrom tlie index of any direct power; or, in other words, divide
t.liftt power successively by its root ; thus, let a" be the direct nS^
pcwerofo, then
«— ' =
a'
a'
a»' =
T
a»' =
^ = . = .
«'' =
°'='='.
a a
1 1 1 1
a <r
"* a? • "* a* "^ '
^, generally, tf~* = — . TheU is^ the value of an inverse potg>er
Septal to unitff divided by the same direct power.
As the properties of inverse powers are the same as those of direct
Jowen, all that has been stated in Section V. regarding the latter,
^oally applies to the former, and it is therefore unnecessary to
recapitulate it here.
We have, however, yet to mention another kind of exponent, viz.,
the fractional, used to express the roots of quantiUes, in a similar
manner to that employed for expressing powers ; and which extends
to them also, all the properties already stated as belonging to inverse
tnd direct powers.
This method of expressing any root of a quantity, consists in
ittaching to that quantity as an exponent, a fraction having unity for
its numerator, and the index of the root for its denominator.
i
90 LOGARITHMS. [pABT I.
Thus \/a is written or
1
i_
And the roots thus expressed may be employed in algebraical cal
culations in the same manner as powers with integral exponents.
Thus, if we wish to express the square of the cube of jr, we do so by
x'^ ^  = a/* ; and in like manner, to express the square of the cube
root of ;r, we should have j?l><* =: x'; or, the square of the square
root of X =. A^^ z:^ xi =^ X itself, which is obvious. And in
It
general, x" may be employed to express the n^ power of the ni^^
root of J*, or what is the same thing, the m^^ root of the n*^ power
of X.
If the fractional exponent has a negative sign, its value will be
found by the rule already given for integral inverse powers; thus,
" 1 1
«'" = — = ;;;— r^ = Unity dividcd bv the m}^ root of the w'**
power of a.
It only remains to state, that in using fractional exponents, we
may substitute any equivalent fraction, or a decimal, in place of the
original exponent; thus a, a*, a**, a*^, are only so many different
ways of expressing the square root of a; advantage may frequently
be taken of this circumstance, to facilitate the working with surds.
By way of recapitulation, then, there are four different kinds of
exponents employed in analytical operations, viz. : —
The posittce iyitegral exponent, as .r", which denotes the direct nth
power of a?, and is equal to x multiplied w times by itself.
The ne(fative integral exponent, as a; " ", which denotes the inverse
n^^ power of ar, and is equal to unity divided n times by x.
The positive fractional exponent, as «", which denotes the direct
n*** root of ;r, and is equal to a quantity, which being multiplied
n times into itself will equal x.
The negative fractional exponent, as x~ » , which denotes the in
verse n^^ root of a;, and is equal to unity divided by the direct
n^ root of X,
Sect. XIII. Logaritfnns.
LooARiTHiis are a series of numbers in arithmetical progression,
answering to another series in geometrical progression ; so ti^en that
in the former corresponds with 1 in the latter.
CHAP. II.] LOGARITHMS. 91
Thus, 0, 1, 2, 3, 4, 5, are the Logs, or aritkmetieai series ;
and 1, 2, 4, 8, 16, 32,  ^'^ ^^^ ^^f ' ^' 9e(nnetricd series, an
' ' ' ' ' 'J swenng thereto.
Or, 0, I, 2, 3, 4, 5, the logarithms ;
and J, 5, 25, 125, 025, 3125, the corresponding numbers.
Or, 0, 1, 2, 3, 4, 5, the logarithms ;
and 1, 10, 100, 1000, 10000, 100000, the corresponding numbers.
In which it will be seen, that by altering the common ratio of the
geometrical series, the same arithmetical series may be made to serve
as the logarithms of any series of numbers. As above, where the
common ratios of the several geometrical series are 2, 5, and 10, re
spectively.
Or, the logarithms of a number may be considered as the indices
or exponents of the powers of some root or radix, which when in
volved to the power indicated by the logarithm, is then equal to the
number. Thus, taking the foregoing series as an illustration, in
which the roots or radices are 2, 5, and 10, respectively, we have
Nos. Logs, Nos. Logs. Nos. JLog».
1 = 2 « i = 5 " 1 = 10 "
2 = 2' 5 = 6* 10 = 10 *
4 = 2 =* 25 = 5 * 100 = 10 «
8 = 2'* 126 = 5 •'* 1000 = 10 »
16 = 2 ^ 625 = 5 ^ 10000 = 10 *
32 = 2 * 3126 = 5 * 1(M)000 = 10 *
in which the exponents in the columns headed Xo^«., are the logar
rithms of the numbers in the first column headed Nos.
The last of these series, or that in which the common ratio of
the geometrical series, or (what is the same thing) the radix of the
system of logarithms is 1 0, is that usually employed for the purpose
of facilitating calculations ; and is the same as the series of logarithms
contained in the Logarithmic Tables in the Appendix to this work.
As in this system of logarithms the numbers whose logarithms are
integral increase as the powers of 10, and it is requisite in calcula
tions to use the logarithms of the intermediate numbers, it becomes
necessary to interpolate with fractional logarithms ; thus the series
then becomes
Nos. Logi.
Nos. Loot.
1 = 10 ••
8 = 10 •*>**
2 = 10 •«"«
9 = 10 •«*«»
3 = 10 *"^'""
10 = 10 »•
4 = 10 •«»•
11 = 10 «•««««
6 = 10 «■«»»'
12 = 10 »•»»•«
6 = 10 *"»»^»
IS = 10 ••'«»"
7 = 10 "^^^
&c. &c.
in which the exponents of the powers of 10, answering to any given
uamber, will be found to correspond with the logarithm of that num
ber as given in Table H. in the Appendix.
92 LOGARITHMS. [PABT I.
In order to explain the properties oF logarithms, and the reason of
the rules given in the section on Logarithmic Arithmetic, let as as
sume any series of numhers, N, N^, Ng, N^, &c., to which corre
spond the logarithms, /, l^y Ly l^y &c., to the same root or radix (r).
1. Then we have, N = r*, Nj = r'*, N^ = r\ &c. ; and by the
rule for multiplpng powers, (page 66,) r'xr'» = r'^'>=Nx N,;
and N.Nj .Ng =r'+'» + '«. Or the logarithm of the product oj
two or more numbers^ is equal to the sum of the logarithms of those
numbers; and conversefyy the logarithm of the quotient of two num^
berSj is equal to the difference of the loaarithms of the dividend and
divisor ; which correspond with the rules given at page 45, for the
multiplication and division of numbers by logarithms.
2. IfN = Nj =No = N3; then we have, by the foregoins,
log (N.N.N .N)= log (N') = r'+ '+' + ' = r*'; and in genenU,
log (N") = r"'. Or the logarithm of the n** power of any quaniitj/ is
equal to n times the logarWim of thai quantity^ whether n is int^al,
fractional^ negative^ or positive.
For let A be the quantity, and let n be negative ; then A"" = — ;,
and log ( — ) = ^ogof I ^ n (log A) ; and as log of 1 =: 0,
log 1 — » log A = — » (log A) = log ( A").
P 
If n = — ; then, let A ? = K ; raising both sides to the power ^,
we have A'' = K^, and their logarithms, p (log A) = ^ (log K) ; then
P 
dividing both sides by 7, we obtain  (log A) = log K = log (Al).
p t I L
Lastly, if » = — , then will A « = — ; and since log (A •) =a
^ a5
(log A) .. log/i \= logl ^ (logA)=  ^ (logA) =
log(A0
8. In Section XI., page 41, it was stated that the decimal portion
of a loearithm was always the same for the same digits, and that ii
was only the characteristic which was altered, by the multiplication
or division of the quantitv by 10, or any power of 10 ; the reason of
this is obvious, since all the powers of 10 have integers only foi
their logarithms.
4. It was also stated, (page 41,) that it was only the characteristic
of the logarithm of a firaction, which was negative, the decimal
portion being always positive; to illustrate this, let us take the
fraction 05 = ^ = — j— = lO"* x lO**"*^** = 10  2+ cwwo .
and therefore 2698970 is the log. of 05.
5. Suppose there be two systems of logarithms whose roots or
CHIP. II.] LOGARITHMS. 93
bises are r and «. Let any number N have p for its loearithm in the
first system, and q for its logarithm in the second : we shidl have N = r^
and N = I* ; which gives »* = #», and * = r». Therefore, taking
the logarithms for the system r, we shall have log 9 =^(logr); or,
if in the system r we have log r = 1, then log » = , or y =
pi
l£ =s p X . . Thus, knowing the logarithm p of any number
N, for the system whose base is r, we may obtain the logarithm q of
tlie same number for the system «, by multiplying je? by a fraction
vlioee numerator is unity and denominator the logarithm of 8 taken
Id the sjstem r.
6 In the system of logarithms first constructed by Baron Napier,
the great inventor, 9 = 2*718281828459, &c., and the exponents
ire usually denominated Napierian^ or Hyperbolic logarithms; the
latter name beins given because of the relation between these log
triihms and the lines and asymptotic spaces in the equilateral hyper
bola: 80 that in this system n is always the hyperbolic logarithm of
(2718*28, &c.)*. But in the system constructed by Mr. Briggs
(correspondm^ with the spaces in a hyperbola whose asymptotes
make an angle of 25° 44' 25" 28'"), called common or Briggean
logarithms, r = 10 ; so that the common logarithm of any number
is, as already stated, the index of that power of 10 which is equal to
the said number.
7. Although the Briggean logarithms are those usually employed
for the purpose of computation, the hyperbolic are always used in
the differentia] calculus, and the higher branches of analysis ; it there
fore becomes somedmes requisite to find the hyperbolic logarithm of
a quantity, which may be obtained from the common logarithm as
follows : from § 5, above, we have q =: p . ; then, * putting
^' log for hyperbolic logarithm, log for common logarithm, and sub
«itating for * its proper value, we have h. log n = log n . ^717757^^7
log 2'71o281
= log n Q.^3^^g^^Q = 23025851 (log «) ; or io obtain the hyper
Ik^Hc logarithm of any number^ multiply its conmion logarithm by
2S025851.
In practice the following method will be found more convenient
tlam multiplying by 23026851.
To convert common logs, into hyperbolic.
Write the common log, as shown in the following Examples, and then
^e from Table I. the equivalent value of each figure 10 hyperbolic
1^ taking care that the latter are each moved as many places to the
'ight as the corresponding numbers in the common logarithm. The
8nm of the whole will then be the hyperbolic logarithm required.
1)4
LOGARITHMS.
[part I.
To convert hypeihdic logs, into common.
Proceed in the same manner, only using Table II. instead of
Table I.
TABLE I.
TABLE XL
S: «yp^^»
Hyp.
Logs.
Com. Logs.
1 23025851
2 46051702
3 , 69077553
4 9 2103404
5 1 115129255
6 ' 138155106
7 161180957
8 , 184206807
9 1 207232658
1
2
3
4
5
6
7
8
9^
•4342945
•8685890
13028834
17371779
21714724
26057669
30400614
34743559
39086503
Examples.
I. What is the hyp. log. of 1662 ?
By reference to Table III. in the Appendix, we find that 3220631
is the common log. of 1662 ; then
7415778?
Cora. Log.
Hyp. Log.
3000000
=
6907755
3
•200000
=
460517
•020000
=
•046051
7
000600
=
•001381
5
000030
=
000069
•000001
log. (
•000002
7415778
3
3220631
i common
)f the numb
er
Hyp. Log.
Com. Log
,
7000000
^
3040061
4
•400000
=
173717
7
•010000
=
•004342
9
005000
^
•002171
4
•000700
^
000304
000070
=
•000030
4
000008
=
•000003
4
7416778
3220631
CHAP. II.] COMPUTATION OP PORMULiG. 95
Sect. XIV. ComptUcUion of FormulcB.
Since the comprehension, and the numerical computation of for
iiu]« expressed aJgebraically, are of the utmost consequence to prac
ical men, enabling them to avail themselves advantageously of the
Lbcorelical results of men of science, as well as to express in scien
tific language the results of their own experimental or other re
semrthes; it has appeared expedient to present brief treatises of
Aiitbinetic and Algebra. The thorough understanding of these two
initiatory departments of science will serve essentially in the applica
QOD of all that follows in the present volume ; and that application
mar probably be fieusilitated by a few examples, as helow : —
£i. I. Let 6 = 5, <r = 12, d = 13, and s = ; then what
Kthe numerical value of the expression \/« («— ^) • (* — c) . (« — </)
from page 1 27, which denotes the area of the triangle whose sides are
a, 12, and 18?
„ 6hc + J 5 + 12 + 13 ,^,, .
Here i = — ^^^^ — = — ■ — ^ = 15 ; then « — ^ = 15
5 = 5; <~c= 15 — 12 = 3; and« — </= 15 — 13 = 2.
CoDseqaently, by substituting the numerical values of the several
qoMtities between the parentheses for them, we shall have
v/(l5 X 10 X 3 X 2) = x/900 = 30, the value required.
The same values being given to 6, c, and c?, we may, as a verifica
tiOQ of the above, compute the area of the same triangle from the
eqaiTalent expression, /\J ^  ( ^^ "^ 2/ ' 2*
Herer = 12 = 144 ; 6 = 5 = 25 ; 2rf = 26; and ^ = 6.
Sabstituung these, the expression becomes
the same result as before.
^J?. 2. Suppose ^ = 32^, < = 6 : required the value of J g f,
•D expression denoting the space in feet which a heavy body would
Wl TerticaUy from quiescence in six seconds, in the latitude of
I<ondon.
Here ^^ = 16^^ x 6* = 96i x 6 = 579 feet.
^3. Given D = 6, rf = 4, A = 12, a = 3*141593 ; required the
»iloeof y^ «^A(D' + D</ + d% a theorem for the solid content of
* conic fhwtam whose diameters of the two ends are D, </, and
lieigbtA.
9fi COMPUTATION OF FORMULA. [PART I.
Here D = 36, D r/ = 6 x 4 = 24, d' =16,^ = 2618 nearly.
Hence jV v A (D* + D<f + d') = 2618 (36 + 24 f 16) 12 =
•2618 X 76 X 12 = 3141593 x 76 = 238761068.
Ex, 4. Let a = I, A = 25, ^ = 193 inches : what is the value of
2 a ^g h ? This heing the expression for the cuhic inches of water
discharged in a second, from an orifice whose area is a, and depth
below the upper surface of water in the vessel, or reservoir, A, both
in inches.
Here2av^^A=:2 v^(25 x 193) = 10 >/193 = 10 x 1389244
= 138*9244 cubic inches.
Ex, 5. Suppose the velocity of the wind to be known in miles per
hour ; required short approximative expressions for the yards per
minute, and for the feet per second.
First 1760 ^ 60 = *y^ = 29 J = 30 nearly.
Also 5280 h (60 x 60) = f^jo = gg = 4^ = 1 J nearly.
If, therefore, n denote the number of miles per hour :
30 n will express the yards per minute; and 1«, the feet per
second.
These are approximative results : to render them correct, where
complete accuracy is required, subtract from each result its 45th part,
or the Ji/lh part of its ninth part.
Thus, suppose the wind blows at the rate of 20 miles per hour :
Tlien 30 n = 30 X 20 = 600 yards per minute, or more cor
rectly 600 — Y:^ = 600 — 13i = 6865 yards.
Also 1 J w = 30 feet per second ;
or, correctly 30 — ^ J = 30 — ^ = 29^ feet
Conversely, f of the feet per second will indicate the miles per
hour, correct within the 45th part, which is to be added to obtain the
true result.
Ex. 6. To find a theorem by means of which it may be ascertained
when a general law exists, and what that law is.
Suppose, for example, it were required to determine the law
which prevailed between the resistances of bodies moving in the air
and other resisting media, and the velocities with which they move.
Let V,, V.^, denote any two velocities, and R,, R.,, the corresponding
resistances experienced by a body moving with tliose velocities : we
wish to ascertain what power of V , it is to which R j is propor
tional. Let X denote the index or exponent of the power : then will
V* : Vlj : : Rj : R^, if a law subsist.
CHAP. II.] COMPUTATION OF FORMULiE. 97
(V \' R
— I : : 1 : — ^ .
(Vo\' R
— ^ I = — ^. This, expressed logarithmically, gives
* X log^=log^;
^^^^logR^__logR,
logV,  logV/
Hence the quotient of the difTerences of the logs, of the resistances,
divided hy the difference of the corresponding velocities, virill express
the exponent z required.
This theorem is of very frequent application in reference to the
motion of cannon balls, of barges on canals, of carriages on rail
roads, &c., and may indeed be applied to the planetary motions.
When two or more values of any quantities occur in a formula, it
is usual to denote both of them by the same letter, distinguishing
between them by the attachment of a small number below it, as in
the last example above, where the two velocities are both denoted by
the letter V, but distinguished by the numerals, as Vj and V^, and
the two resistances by R^ and R^.
08 OKOMETRICAL DBF1NITI0N8. [PART I.
CHAP. III.
PLANE AND SOLID GEOMETRY.
Sect. I. Defi?iitions.
I. Geometry is that departmcDt of science, by means of which
we (Icnionstrate the properties, affections, and measures of all sorts of
ma^7iitude,
ii. Magnitude is a term used to denote the extension of any thing,
and is of three kinds ; as the magnitude of a /twe, which is only in
one direction, viz. length ; the magnitude of a surface^ which is in tvo
directions, viz. length and breadth; and the magnitude of a solid^
which is in three directions, viz. lengthy breadth and depth,
3. A ]X)int has no parts or magnitude ; neither length, breadth, nor
thickness, and serves only to assign position,
4. A line has length without breadtli or thickness, and indicates
direction and distance.
Cor, The extremities of a line are, therefore, points.
5. A right line is that which lies evenly, or in the same direction,
between two points. A curve liyie continually changes its direction.
Cor, Hence there can only be one species of right lines, but there
is an infinite variety in the species of curves.
6. An angle is the inclination of two lines to one another, meeting
in a point, called the angular point When it is formed by two rig^
lines^ it is a plane angle, as A ; if by curve lines, it is a curvUineal
angle, as B. (Fig. 1.)
7. A right angle is that which is made by one right line A B fall
ing upon another C D, and making the angles on each side equal,
that is, A B C = A B D ; so that the line A B does not incline more
to one side than another : it is then said to be perpendicular to the
line C D. All other angles are called oblique angles. (Fig. ^.)
8. An obtuse angk \s greater than a right angle, as R. (Fig. 3.)
0. An acute angle is less than a right angle, as S. (Fig. 3.)
10. Contiguous or adjacent angles^ are the two angles formed by
one line falling upon another, as R, S. (Fig. 3.)
I I . Vertical or opposite angles, are those made on contrary sides
of two lines intersecting one another, as A £ C, D £ B. (Fig. 4.)
12. A surface has length and breadth, but no thickness, and shows
extension. The area of a figure is the quantity of space which its
surface occupies.
CHAP. III.] ANGLES, AND BIGHT LINES. 99
Cor, The boundaries or limits of a surface are lines.
13. A plane is that surface which lies perfectly even between its
extremes; or which, being cut by another plane in any direction, its
section would be a straight line.
14. A idid is a magnitude extended every way, or which has
length, breadth, and depth.
Cor, The boundaries or extremes of a solid are surfaces.
15. The square of a right line is the space included by four right
lines equal to it, set perpendicular to one another.
16. The rectangle of two lines is the space included by four lines
equal to them, set perpendicular to one another, the opposite ones
being equal.
17. One right line is said to he parallel io another, when both lines,
being extended to any length, will never meet, but always preserve
the same distance between them.
Sect. II. Of Angles^ and Right Lines^ and their Rectangles,
Prop. I. If to any point C in a right line A B, several other right
lines D C, £ C are drawn on the same side ; all the angles formed at
the point C, taken together, are equal to two right angles ; thus,
ACD + DCE + ECB = the two right angles A C P + P C B.
(F%. 5.) rEuclid, Book I. Prop. 13.]
Cor, 1. AH the angles made about one point in a plane, being
taken together, are equal to four right angles.
Cor. 2. If all the angles at C, on one side of the line A B, are
found to be equal to two right ancles ; then A C B is a straight line.
(Fig. 5.) [Euc. B. I. Prop. 14.J
II. If two right lines, A B, CD, cut one another, the opposite
angles C £ B and A £ D will be e^ual. (Fig. 4.) [Euc. B. I. Prop. 15.]
III. A right line, B I, which is perpendicular to one of two pa
rallels, is perpendicular to the other. (Fig. 6.)
IV. If a right line C G, intersects two parallels AD, F H ; the
alternate angles, ABE, and B £ H, will be equal. (Fig. 6.) QEuc.
B. I. Prop. 29.]
Cor. 1. The two internal angles D B £ and B E H on the same side
are equal to two right angles. (Fig. 6.)
Cot. d. The external angle C B D, is equal to the internal angle on
the same side B £ H. ^Fig. 6.) [Euc. B. I. Prop. 28.]
V. Right lines, parallel to the same right line, are parallel to one
another. [Euc. B. I. Prop. 80.]
VI. If a right line A G be divided into two parts A B, B C ; the
square of the whole line is equal to the squares of both the parts, and
twice the rectangle of the parts ; or A C* = A B* + B C'^ + 2 A B x
BC. (Pig. 7.) [Euc. B. II. Prop. 4.]
VII. The square of the difference of two lines A C, B C, is equal
to the sum of their squares, wanting twice their rectangle ; or A B^ ^
AC" + BC»— 2AC X BC. OrAB* f 2AC X BC = AC* f
B C*. (Fig. 7.) [Euc. B. II. Prop. 7.]
H 2
100 TRIANGLES. [PABT I.
VIII. The square of the sum of two lines is equal to the sum of
their squares, together with their rectangle.
IX. The rectangle of the sum and difference of two Hoes is equal
to the difference of their squares.
X. The square of the sura, together with the square of the differ
ence of two lines, is equal to twice the sum of their squares.
Sect. III. Of Triangles.
DEFINITIONS.
1 . A triangle is a plane figure bounded by three right lines^ called
the sides of the triangle.
2. An equilateral triangle is one in which all the three sides are
equal.
3. An isosceles triangle has only two sides equal.
4. A scalene triangle has all its three sides unequal.
5. An equiangular triangle is one which has three equal angles;
and two triangles are said to be equiangular, when the angles in the
one are respectively equal to those in the other.
6. A rightanghd triangle is that which has one of its angles a right
angle. The side opposite to the right angle is called the hypotenuse^
and the other two sides the legs.
7. An oUique angled triangle has all its angles oblique.
8. An obtuse angled triangle has one of its angles obtuse.
9. An acute angled triangle has all its angles acute.
10. In a triangle, the lowest side, as A B, is called the hase^ and
the opposite angle C the vertex : the altitude of a triangle is the per
pendicular height (CD) from the base, or its extension to the yertex.
(Fig. 8.)
1 1. Similar triangles are those whose angles or sides are Tesped^
ively equal, each to each. And homologous sides are those lying
between equal angles.
PROPOSITIONS.
Prop. I. In any triangle ABC, if one side B C be produced or
drawn out ; the external angle A C D will be equal to the two inter
nal opposite angles ABC and BA C. (Fig. 9.) [Euc. B.I. Prop. 32.1
II. In any triangle, the sum of the three internal angles is equal
to two right angles. [Euc. B. I. Prop. 32.]
Cor, 1. If two angles in one triangle be equal to two angles in
another : the third will also be equal to the third.
Cor, 2. If one angle of a triangle be a right angle, the sum of the
other two will be equal to a right angle.
III. The angles at the base of an isosceles triangle, are equal.
[Eiic. B. I. Prop. 5.]
Cor, 1 . An equilateral triangle is also equiangular ; and the con
trary.
Cor. 2. The line which is perpendicular to the base of an iaosceles
triangle, bisects both it and the yertical angle.
CHIP, in.] TRIANGLES. 101
IV. In any triaogle, the greatest side is opposite to the greatest
ugle, and the least to the least. [Euc. B. I. Prop. 18.]
V. In any triangle ABC, the sura of any two sides B A, A C, is
greater than the third B C, and their difTerence is less than the third
ade. (Fig. 10.) [Euc. B. I. Prop. 20.]
VI. If two triangles ABC, a 6 c, have two sides, and the included
togle equal in each ; these triangles, and their correspondent parts,
M be equal. (Fig. 11.) [Euc. B. I. Prop. 4.]
VII. If two triangles ABC and abc^ have two angles and an in
cloded side equal, each to each ; the remaining parts shall he equal,
•ndthe whole triangles equal. (Fig. 11.) [Euc. B. I. Prop. 26.]
VIII. Triangles of equal bases and heights are equal. [Euc. B. I.
Prop. 37.]
IX. Triangles of the same height, are in proportion to one another
as their bases. [Euc. B. VI. Prop. 1.]
X. If a line D E be drawn parallel to one side B C, of a triangle ;
the segments of the other sides will be proportional ; that is,
AD:DB:: AE : EC. (Fig. 12.) [Euc. B. VI. Prop. 2.]
Cor, 1. If the segments be proportional, A D : D B : : A E : EC ;
then the line D £ is parallel to the side B C. (Fig. 12.)
C<fr. 2. If several lines be drawn parallel to one side of a triangle,
>I] the segments will be proportional.
Cor. 3. A line drawn parallel to any side of a triangle, cuts off a
triiDgle similar to the whole.
XI. Id similar triangles, the homologous sides are proportional ;
thatis,AB : AC :: DE : DF. (Fig. 13.) [Euc. B. VI. Prop. 4.]
XII. Like triangles are in the duplicate ratio, or as the squares of,
their homologous sides. [Euc. B. VI. Prop. 19 ]
XIII. In any triangle, the difference of the squares of the two legs
A Band A C, is equal to twice the rectangle contained by the base
Be, and the distance DO, of its middle point from the perpendicu
lar DA. (Fig. 14.)
XIV. In a rightangled triangle B A C, if a perpendicular be let
fill from the right angle upon the hypothenuse, it will divide it into
two triangles, similar to one another and to the whole, A B D, A D C.
(rig.l4.) [Euc. B. VI. Prop. 8.]
Cor, 1. The rectangle of the hypothenuse and either segment is
eqoal to the square of the adjoining side.
XV. The distance A O of the right angle, from the middle of the
hjpotbenuse is equal to half the hypothenuse. (Fig. 14.)
XVI. In a rightangled triangle, the square of the hypothenuse is
eqwd to the sum of the squares of the two sides. [Euc. B. I. Prop. 47.]
XVII. If the square of one side of a triangle be equal to the
•om of the squares of the other two sides ; then the angle compre
^ed by them is a right angle. [Euc. B. I. Prop. 48.]
XVIII. I fan angle A, of a triangle B A C be bisected by a right
be A D, which cuts the base ; the segments of the base will be
Proportional to the adjoining sides of the triangle; that is,
BD: DC :: AB : AC. (Fig. 10.) [Euc. B. VI. Prop. 3.]
102 QUADRILATERALS AND POLYGONS. [PABT I.
XIX. If the sides be as the segments of the base, the line A D
bisects the angle A. (Fig. 10.) [Euc. B. VI. Prop. A.l
XX. Three lines drauTi from the three angles of a triangle to the
middle of the opposite sides, all meet in one point.
XXI. Three perpendicular lines erected on the middle of the three
sides of any triangle, all meet in one point.
XXII. The point of intersection of the three perpendiculars, will
be equally distant from the three angles ; or, it will be the centre of
the circumscribing circle.
XXIII. Three perpendiculars drawn from the three angles of a
triangle, upon the opposite sides, all meet in one point.
XXIV. Three lines bisecting the tliree angles of a triangle, all
meet in one point.
XXV. If D be any point in the base of a scalene triangle, ABC:
then is A B X D C f A C x B D = AD^ xBCfBCxBD
X DC. (Fig. 10.)
Sect. IV. Of QuadrUatercdB and Pdi/gons,
DEFINITIONS.
1. A quadrangle or quadrilaieral^ is a plane figure bounded by four
right lines.
2. A paraUelogram is a quadrangle whose opposite sides are pa
rallel, as AGBH. The line A B drawn to the opposite corners is
called the diameter or diagonal. And if two lines be drawn parallel
to the two sides, through any point of the diagonal, they divide it
into several others, and then C and D are called parallelograms about
the diameter; and E and F the complements; and the figure £ DF a
gnomon, (Fig. 15.)
3. A rectangle is a parallelogram whose sides are perpendicular to
one another.
4. A square is a rectangle of four equal sides and four equal
angles.
5. A rhombus is a parallelogram, whose sides are equal, and all its
angles oblique.
6. A rhomboid is a parallelogram in which only the opposite sides
arc equal, and all its angles oblique.
7. A trapezoid is a quadrangle, having only two sides parallel, as
Fig. IG.
8. A trapeziuyn is a quadrangle that has no two sides parallel,
as Fig. 17.
9. A pdygon is a plane figure enclosed by many right lines. If
all the sides and angles are equal, it is called a regular polygon, and
denominated according to the number of sides or angles, as a pen
tagon having five sides, a hexagon^ having six sides, a heptagon^ having
seven sides, &c.
10. The diagonal of a quadrangle or polygon is a line drawn be
tween any two opposite comers of the figure, as A B. (Fig. 15.)
diP.ni.] QUADBI LATERALS AND POLYGONS. 103
11. The height of a figure is a line drawn from its vertex^ perpen
^licular to the base* or opposite side on which it stands.
12. Like or nmiiar figures, are those whose several angles are
•equal to one another, and the sides about the equal angles proper
lioQiL
13. Homelogmte sides of two like figures are those between two
angles, respectively equal.
14. The perimeter or circumference of a figure, is the compass of
it, or sum of all the lines that enclose it.
15. The internal angles of a figure are those on the inside, made
l»y the lines that bound the figure, as A B C, D C B, &c. (Fig. 18.)
16. The external angles of a figure are the angles made by each
lide of a figure, and the adjoining side drawn out, as B A F, A £ G.
(Fig. 18.)
PROPOSITIONS.
Pbop. I. In any parallelogram the opposite sides and angles are
equal ; and the diagonal divides it into two equal triangles. [Euc.
B. I. Prop. 34.]
II. The diagonals of a parallelogram intersect each other in the
middle point of both.
III. Any line B C passing through the middle of the diagonal of
» parallelogram P, divides the area into two equal parts. (Fig. 19.)
IV. Any right line B C drawn through the middle point P of the
diigooal of a parallelogram, is bisected in that point ; or B P = P C.
V. In any parallelogram A G H B, the complements £ and F are
equal. (Fig. 15,) [Euc. B. I. Prop. 43. J
VI. Parallelograms of equal bases and heights are equal. [Euc.
B. I. Prop. 36.]
VII. The area of a parallelogram is double the area of a triangle
binngthe same base and height. [Euc. B. I. Prop. 41.]
VIII. Parallelograms of the same height are to one another as
tbeir bases. [Euc. B. VI. Prop. 1.]
IX. Parallelograms of equal bases are as their heights.
X. Parallelograms are to one another, as their bases and heights.
XI. In any parallelogram the sum of the squares of the diagonals
ii eqoal to the sum of the squares of all the four sides.
XII. The sum of the four internal angles of any quadrilateral
%ore, is equal to four right angles.
XIII. If two angles of a quadrangle be right angles, the sum of
tlie other two amounts to two right angles.
XIV. The sum of all the internal angles of a polygon is equal to
tvice as many risht angles, abating four, as the polygon has sides.
Cor. Hence all rightlined figures of the same number of sides,
hate the sum of all the internal angles equal.
Xy. The sum of the external angles of any polygon is equal to
four right angles.
Cor, All rightlined figures have the sum of their external angles
eqoal.
104 CIRCLBP, ETC. [part I.
XVI. In two similar figures AC, PR; if two lines BE, Q T, be
drawn after a like manner, as suppose, to make the angle C B E =
R Q T ; then these lines liave the same proportion as any two
homologous sides of the figure ; viz.,
BE:QT::BC:QR::AB:PQ::AD:PS. (Fig. 20.)
XVII. All similar figures are to one another as the squares of
their homologous sides.
XVIII. Any figure described on the hypothemise of a right
angled triangle, is equal to two similar figures described the same
way upon the two legs; that is, BFC=ALC fAGB. (Fig. 21.)
[Euc. B. VI. Prop. 31.]
XIX. Any regular figure ABCDE, is equal to a triangle whose
base is the perimeter A B C D E A ; and height the line O P, drawn
from the centre, perpendicular to one side. (Fi<r. 22.)
XX. Only three sorts of regular figures can fill up a plane surface,
that is, the whole space round an assumed point, and these are six
triangles, four squares, or three hexagons.
Sect. V. Of the Circle^ and Inscribed and Circumscribed Figures,
DEFINITIONS.
1. A circle is a plane figure described by a right line moving
about a fixed ]>oint, as A C about C : or it is a figure bounded by a
curved line, every part of which is equidistant from a fixed point.
(Fig. 23.)
2. The centre of a circle is the fixed point about which the line
moves, as C. (Fig. 23.)
3. The radius is the line that describes the circle, as C A. (Fig. 23.)
Cor, All the radii of a circle are equal.
4. The circumference is the line described by the extreme end of
the moving line, as A B D E A. (Fig. 23.)
5. The diameter is a line drawn through the centre, from one side
to the other, as A D. (Fig. 23.)
6. A semicircle is half the circle, cut off by the diameter, as ABD.
7. A quadrant,, or quarter of a circle, is the part between two
radii perpendicular to one another, as C DE. (Fig. 23.)
8. An arc is any part of the circumference, as A B. (Fig. 24.)
.9. A sector is a part bounded by two radii, and the arc between
them, as A C B. (Fig. 24.)
10. A chord is a right line drawn through the circle, as D F.
1 1. A segment is a part cut off by a right line, or chord, as D £ F,
or DABfI (Fig. 24.)
12. Angle at the centre is that whose angular point is at the centre
A C B. (Fig. 24.)
13. Angle at the circumference is when the angular point is in the
circumference, as BAD, or BCD. (Fig. 25.)
14. Angle in a segment^ is the angle made by two lines drawn
CHAP. III.] CIRCLES, ETC. 105
irom some point of tbe arc of that segment to the ends of the base ;
IS BCD is an angle in the segment BCD. (Fig. 25.)
15. Angle upon a segment is the angle made in the opposite
Kgment, whose sides stand upon the base of the first ; as BAD,
which stands upon tlie segment BCD. (Fig. 25.)
16. A tangent is a line touching a circle, which, produced, does
sot cat it, as 6 A F. (Fig. 23.)
IT. Circles are said to touch one another, which meet, but do not
eat one another.
18. Similar arcs, or similar sectors^ are those bounded by radii
that make the same angle.
19. Similar segments are those which contain similar triangles,
ilike placed.
20. A figure is said to be inscribed in a circle, or a circle circum
mhed about a figure^ when all the angular points of the figure are
in tbe circumference of the circle. (Fig. 26.)
21. A circle is said to be inscribed in a figure, or a figure circum^
Knhed about a circle, when the circle touches all the sides of the
figure. (Fig. 27.)
22. One figure is inscribed in awo/Aer, nvhen all the angles of the
mscribed figure are in the sides of the other. (Fig. 28.)
PROPOSITIONS.
Phop. I. The radius C R, bisects any chord at right angles, which
does not pass through the centre, as A B. (Fig. 29.) [Euc. B. III.
Prop. 3.]
Cor, 1. If a line bisects a chord at right angles, it passes through
tie ccDtre of the circle.
Cor. 2. The radius that bisects the chord also bisects the arc.
II. In a circle equal chords are equally distant from the centre.
[Edc. B. III. Prop. 14.]
III. If 8e?eral lines be drawn through a circle, the greatest is the
diaineter, and those that are nearest the centre are greater than those
tbtt are farther off. fEuc. B. III. Prop. 15.]
IV. If from any point three equal right lines can be drawn to the
ojcamference, that point is the centre. [Euc. B. III. Prop. 9.]
V. No circle can cut another in more than two points. [Euc.
B. III. Prop. 10.]
yi. There can be only two equal lines drawn from any exterior
point, to the circumference of a circle.
VIL In any circle, if several radii be drawn making equal angles,
tie arcs and sectors comprehended thereby will be equal ; that is, if
tbe angle A C B = B C D, then, the arc A B = B D, and the sector
ACB = BCD. (Fig. 30.) [Euc. B. III. Prop. 26.]
VIII. In the same or equal circles, the arcs, and also the sectors,
ve proportional to the angles intercepted by the radii.
IX. The circumferences of circles are to one another as their
diuneters.
X. A right line, perpendicular to the diameter of a circle, at the
106 CIBCLBS, ETC. [PART I.
extreme point, toncbes die circle in that point, and lies wholly withoot
the circle ; or is a tangent to the circle.
XI. If two circles touch one another, either inwardly or outwardly,
the line passing throogh their centres shall also pass through the
point of contact. [Euc. B. III. Prop 11 and 12.]
XII. In a circle the angle at the centre is double the angle at the
circumference, standing upon the same arc; orBDC = 2BAC
(Fi*r. 31.) Euc. B. III. Prop. 20.]
XIII. All angles in the same segment of a circle are equal, as
DAC = DBC, and DOC=DHC. (Fig 32.) [Eac. B. III.
Prop. 21.;
XIV. if two right lines DC, A B, be drawn from the extremities
of two equal arcs D A, B C, they will be parallel. (Fig. 32.)
XV. The angle A B C in a semicircle is a right angle. (Fig. 33.)
[Euc. B. III. Prop. 31.;
XVI. The angle FBO, in a greater segment FABCO, is less
than a right angle ; and the angle D B E, in a less segment D B E,
is greater than a right angle. (Fig. 33.) [Euc. B. III. Prop. 31.]
X VII. If two lines cutting a circle, intersect one another in A ;
and there be made at the centre, Z.ECF = Z.BAD; then the arc
B D 4 G H = 2 E F, if A is within the circle ; or the arc B D —
G H == 2 E F, if A is without. (Fig. 34.)
XVIII. If from a point without, two lines, A B, AD be drawn
to cut a circle ; the angle made by them is equal to the angle at the
centre, standing on half the difference of the two arcs of the circum
ference G H, B D. (Fig. 31..)
XIX. The angle A = Z.BHDfHDO, when A is witliin ; or
A = B H D — H D G, when A is without the circle. (Fig. 34.)
XX. In a circle, the angle made at the point of contact between
the tangent and any chord, is equal to the angle in the alternate or
opposite segment; ECF=EBC, and ECA=EGC. (Fig. 35.)
[Euc. B. III. Prop. 32.]
XXI. A tangent to the middle point of an arc, is parallel to the
chord of it.
XXII. If from any point B in a semicircle, a perpendicular B D
be let fall upon the diameter, it will be a mean proportional between
the segments of the diameter; that is, AD : DB :: DB : DC.
(Fig. 30.) [Euc. B. VI. Prop. 13.]
XXIII. The chord is a mean proportional between the adjoining
segment and the diameter, from the similarity of the triangles : that
is, A D : A B :: A B : AC ; and C D : CB :: C B : CA. (Fig. 36.)
XXIV. In a circle, if the diameter A D be drawn, and from the
ends of the chords A B, AC, perpendiculars be drawn upon the
diameter ; the squares of the chords will be as the segments of the
diameter ; that is, A E : A F :: A B' : AC". (Fig. 37.)
XXV. If two circles touch one another in P, and the line PD£
be drawn through their centres ; and any line P A B is drawn
through that point to cut the circles, that line 'will be divided in
proportion to the diameters; that is PA : PB :: PD : PE. (Fig. 38.)
CHlP.m.] CIRCLES, ETC. 107
XXVI. If through any point F in the diameter of a circle, any
chord, C F D be drawn, the rectangle of the segments of the chord is
eqaal to the rectangle of the segments of the diameter ; C F x F D
= AF X FB = also GF x FE. (Fig. 39.) [Euc. B. III. Prop. 35.]
XXVII. If through any point F out of the circle in the diameter
BA produced, any line F C D be drawn through the circle : the rect
ingle of the whole line and the external part is equal to the rectangle
of the whole line passing through the centre, and the external part ;
DFxFC = AF X FB = also F E X F G. (Fig. 40.)
XXVIIL Let H F be a tangent at H ; then the rectangle C F x
FD = square of the tangent F H. (Fig. 40.) [Euc. B. III. Prop. 36.]
XXIX. If from the same point F, two tangents be drawn to the
circle, they will be equal ; that is, F H = F I. (Fig. 40.)
XXX. If a line P F C be drawn perpendicular to the diameter
AD of a circle; and any line drawn from A to cut the circle and
the perpendicular ; then the rectangle of the distances of the sections
from Ay will ^be equal to the rectangle of the diameter and the
distance of the perpendicular from A ; that is, A B x A C =
AP X AD. (Fig. 41.)
Also, A B X A C = A K\ (Fig. 41.)
XXXI. In a circle E D F whose centre is C, and radius C E, if
tbc points B, A, be so placed in the diameter produced, that C B,
C£, CA be in continual proportion, then two lines BD, A D drawn
from these points to any point in the circumference of the circle
will always be in the given ratio of BE to A E. (Fig. 42.)
XXXII. In a circle, if a perpendicular D B be let fall from any
point D, upon the diameter C I, and the tangent D O drawn from D,
tken AB, AC, AO, will be in continual proportion. (Fig. 43.)
XXXIII. If a triangle B D F be inscribed in a circle, and a per
pendicular D P let fall from D on the opposite side B F, and the
<iiMieter D A drawn ; then, as the perpendicular is to one side
inclading the angle D, so is the other side to the diameter of the
ciitle; that is, D P : D B :: D F : D A. (Fig. 44.)
XXXI y. The rectangle of any two sides of an inscribed triangle
is equal to the rectangle of the diameter, and the perpendicular on
tl»e third side ; that is, BDxDF=ADxDP. (Fig. 44.)
XXXV. If a triangle B A C be inscribed in a circle, and the angle
A hiflectcd by the right line A E D, then as one side is to the seg
nicDt of the bisecting line within the triangle, so is the whole bisect
ing line to the other side; that is, A B : AE :: AD : A C; and
ABxAC = BE.EC + AE2. (Fig. 45.)
XXXVI. If a quadrilateral A BCD be inscribed in a circle, the
mn of two opposite angles is equal to two right angles; that is,
ADClABC= two right angles. (Fig. 46.) [Euc. B. III.
Prop. 22.]
XXXVII. If a quadrangle be inscribed in a circle, the rectangle
of the diagonals is equal to the sum of the rectangles of the opposite
mde»; or c A X BD = CB x DA + CD x A B. (Fig. 46.)
108 CIRCLES, ETC. [pART I.
XXXVIII. A circle is equal to a triangle whose base is the cir
cumference of the circle ; and height, its radius.
XXXIX. The area of a circle is equal to the rectangle of half the
circumference and half the diameter.
XL. Circles (that is, their areas) are to one another as the squares
of their diameters, or as the squares of the radii, or as the squares of
the circumferences. [Euc. B. XII. Prop. 2.]
XLI. Similar polygons inscribed in circles, are to one another as
the circles wherein they are inscribed.
XLII. A circle is to any circumscribed rectilineal figure, as the
circle's periphery to the periphery of the figure.
XLIII. If an equilateral triangle ABC be inscribed in a circle ;
the square of the side thereof is equal to three times the square of the
radius ; that is, A B* = 3 A D^. (Fig. 47.)
XLIV. If from any point D in the circumference of a circle,
having inscribed in it an equilateral triangle, chords be drawn to the
three angles A, B, C ; the longest chord A D, is equal to the sum of
the two lesser chords, B D and C D. (Fig. 48.)
XLV. A square inscribed in a circle, is equal to twice the square
of the radius.
XLVI. The side of a regular hexagon inscribed in a circle, is
equal to the radius of the circle.
XLVII. If two chords in a circle mutually intersect at right
angles, the sum of the squares of the segments of the chords is equal
to the square of the diameter of the circle ; that is, A P^ f P B* r
P C*^ h P D' = diam.^. (Fig. 49.)
XLVIII. If the diameter P Q be divided into two parts at any
point R, and if R S be dmwn per]>endicular to P Q ; also R T ap
plied equal to the radius, and T R produced to the circumference
at V: then,
PR + RQ ., ., ., \
R T = ; or R T is the arithmehcal mean, ] between the
f two segments
RS = n/pR X RQ; or RS is the geometrical me&n. ;of the diame
« ^ « inT^ ' I ter P R, R Q
RV = : or RV is the Aar mow iW mean. ^ ^^^f^' ^
PR + RQ
XLIX. If the arcs P Q, QR, RS, &c., be equal, and there be
drawn the chords P Q, PR, PS, &:c., then it will be P Q : P R ::
PR : PQ f PS :: PS : PR f PT :: PT : PS f P V, &c. (Fig. 51.)
L. If the arcs PQ, Q R, R S, &c. be equal, the angles Q P R, RP8,
SPT,&c., will be equal ; or in equal circles, equal angles stand upon
equal arcs, whether they be at the centres or circumferences. (Fig.
51.) [Euc. B. III. Prop. 20.]
LI. The centre of a circle being O, and P a point in the radius,
or in the radius produced ; if the circumference be divided into as
many equal parts A B, B C, C D, &c., as there are units in 2 n, and
lines be drawn from P to all the points of division ; then shall the
continual product of all the alternate lines, viz. PAx PC x PE, &c..
CHIP. III.] PLANES AND SOLIDS. 109
be = r" — x" when P is within the circle, or = a? — r* when P is
without the circle; and the product of the rest of the lines, viz.,
PB X P D X P F, &c., = r" H ;i^ : where r = A O the radius, and
« = 0P the distance of P from the centre. (Fig. 52.)
Sect. VI. Of Planes and Solids.
DEFINITIONS.
1. The common section of two planes, is the line in which they
meet, or cat each other.
2. A line is perpendicular to a plane, when it is perpendicular to
erenr line in that plane which meets it.
3. One plane is perpendicular to another, when every line of the
one, which is perpendicular to the line of their common section, is
perpendicolar to the other.
4. The inclination of one plane to another, or the angle they
fonn between them, is the angle contained by two lines, drawn from
anr point in the common section, and at right angles to the same,
ODC of these lines in each plane.
5. Parallel planes are such as being produced ever so far in any
direction, will never meet, or which are everywhere at an equal per
pendicular distance.
6. A solid an^ is that which is made by three or more plane
tngles, meeting each other in the same point.
7. Similar solids^ contained by plane figures, are such as have all
their solid angles equal, each to each, and are bounded by the same
nomber of similar planes, alike placed.
8. A prism is a solid whose ends are parallel, equal, and like
plane figures, and its sides, connecting those ends, are parallelograms.
(Fig. 53 and 54.)
9. A prism takes particular names according to the figure of its
hue or ends, whether triangular, square, rectangular, pentagonal,
hexagonal, &c.
10. A right or upright prism^ is that which has the planes of the
fldftf perpendicular to the planes of the ends or base. (Fig. 53.)
When such is not the case it is called an oblique prism. (Fig. 54.)
11. A parallelapipedj or parallelopipedon, is a prism bounded by
Bx parallelograms, every opposite two of which are equal, alike, and
parallel. (Fig. 55.)
12. A rectangular parallelopipedon is that whose bounding planes
>re all rectangles, which are perpendicular to each other. (Fig. 56,)
13. A cube is a square prism, being bounded by six equal square
sdea or faces, which are perpendicular to each other. (Fig. 57.)
14. A cylinder is a round prism having circles for its ends ; and is
conceived to be formed by the rotation of a right line about the
cirtufflferences of two eqtuJ and parallel circles, always parallel to
^axis. (Fig. 58.)
110 SOLID GEOMETRY. [PART I.
15. The axis of a cylinder is the right line A B joining the centres
of the two parallel circles, about which the figure is described.
IG. A pyramid is a solid whose base is any rightlined plane
figure, and its sides triangles, having all their vertices meeting
together in a point above the base, called the vertex of the pyramid.
(Fig. 59.)
1 7. Pyramids, like prisms, take particular names from the figure
of their base.
18. A cone is a round pyramid having a circular base, and is con
ceived to be generated by the rotation of a right line about the
circumference of a circle, one end of which is fixed at a point aboYe
the plane of that circle. (Fig. 60.)
1 9. The axis of a cone is the right line, A B, joining the Tertex,
or fixed point, and the centre of the circle about which the figure is
described.
20. When the axis of a cone or pyramid is perpendicular to the
base, it is called a right cone or pyramid ; but if inclined it is called
oblique,
21. Similar cones and cylinders^ are such as have their altitudes
and the diameters of their bases proportional.
22. A sphere is a solid bounded by one curve surface, which is
everywhere equally distant from a certain point within, called the
centre. It is conceived to be generated by the rotation of a semi
circle about its diameter, which remains fixed. (Fig. 61.)
23. The axis of a sphere is the right line about which the semi
circle revolves, and the centre is the same as that of the reyolving
semicircle.
24. The diameter of a sphere is any right line passing through
the centre, and terminated both ways by the surface.
25. The attitude of a solid is the perpendicular drawn from the
vertex to the opposite side or base.
PROPOSITIONS.
Prop. I. If any prism be cut by a plane parallel to its base, the
section will be equal and like to the base.
II. If a cylinder be cut by a plane parallel to its base, the section
will be a circle, equal to the base.
III. All prisms and cylinders, of equal bases and altitudes, are
equal to each other. [Euc. B. XI. Prop. 31.]
IV. Rectangular parallclopipcdons, of equal altitudes, are to each
other as their bases. [Euc. B. XI. Prop. 32.]
V. Rectangular parallelopipedons, of equal bases, are to each
other as their altitudes.
VI. Because prisms and cylinders are as their altitudes, when
their bases are equal : and, as their bases when their altitudes are
equal. Therefore, universally, when neither are equal, they arc to
one another as the product of their bases and altitudes : hence, also,
these products are the proper numeral measures of their quantities or
magnitudes.
CHAP. III.] SOLID GEOMETRY. Ill
VII. Similar prisms and cylinders are to each other as the cubes
of their altitudes, or of any like linear dimensions. [Euc. B. XI.
Prop. 33.]
VIII. In any pyramid a section parallel to the base is similar to
the base ; and these two planes are to each other as the squares of
their distances from the vertex.
IX. In a right cone, any section parallel to the base is a circle ;
and this section is to the base as the squares of their distances from
the vertex.
X. All pyramids and cones of equal bases and altitudes, are equal
to one another.
XI. Every pyramid is the third part of a prism of the same base
and altitude. [Euc. B. XII. Prop. 10.]
XII. If a sphere be cut by a plane, the section will be a circle.
XIII. Every sphere is twothirds of its circumscribing cylinder.
XIV. A cone, hemisphere, and cylinder of the same base and
altitude, are to each other as the numbers 1, 2, 3. [Euc. B. XII.
Prop. 10.]
XV. All spheres are to each other as the cubes of their diameters;
all these being like parts of their circumscribing cylinders. [Euc.
B. XII. Prop. 18.]
XVI. There are only three sorts of regular plane figures which
can be joined together, so as to form a solid angle ; viz. three, four,
or five triangles^ three squares^ and three pentagons: thus giving five
different species of solid angles, which are those of the five regular
or Platonic bodies ; viz. : —
1. The tetraedron^ (^^g* ^2,) bounded by four equilateral tri
angles^ each solid angle of which is formed by three triangles,
2. The kexaedron^ or cuhcy (Fig. 57,) contained by six squares^
each solid angle of which is formed by three squares.
3. The octaedron, (Fig. 63,) bounded by eight triangles^ each
solid angle of which is formed by four triangles,
4. The dodecaedron^ (Fig. 64,) bounded by twelve pentagons^
each solid angle of which is formed by three pentagons,
5. The icosaedron, (Fig. 65,) bounded by twenty triangles, each
solid angle of which is formed by five triangles,
XVII. Only one sort of the foregoing five regular bodies, joined
at their angles, can completely fill a solid space, without leaving any
Tacuity ; viz. eight hexaedrons, or cubes.
XVIII. A sphere is to any circumscribing solid B F, (all whose
planes touch the sphere,) as the surface of the sphere to the surface
of the solid, r Fig. 66.)
XIX. All bo£es cii*cumscribing the same sphere, are to one an
other as their surfaces.
XX. The ^here is the greatest or most capacious of all bodies of
equal surface.
112 PRACTICAL GEOMETRY. [PART I.
Sect. VII. Practical Geometry.
It is not intended in this place to present a complete collection of
Geometrical Problems, but merely a selection of the most useful,
especially in reference to the employments of Mechanics and En
gineers.
The instruments for the purposes of geometrical construction, are
too well known to require any description here; and their use is
much easier learned by an examination of the instruments themselves,
than by any written explanation.
Prob. I. From a given point B in a given straight line A C, to
draw a line perpendicular to the same.
From B as a centre, witli any radius, describe arcs cutting A C in
a and c, then from a and c with any larger radius describe arcs
cutting each other in d and e ; then the straight line d'Be vnW be
perpendicular to A C. (Fig. 6'7.)
Prob. II. To erect a perpendicular at the end of a given line.
With any radius, and from a point somewhere above A B, describe
a semicircular arc passing through the point B, at which the perpen
dicular is to be erected, and also cutting A B in C ; then, through C
and the centre of the arc produce the right line C D, cutting the arc
in D, and a line joining D and B will be the perpendicular required.
(Fig. 68.)
Prob. III. To bisect any given angle A C B.
From C as a centre, with any radius, describe an arc cutting the
sides in D and £ ; then from D and £, as centres with the same
radius, describe arcs cutting each other in F ; then the straight line
joining C and F will bisect the angle A C B. (Fig. 69.)
Prob. IV. To bisect a given angle BAP; then to bisect its half;
and so on.
Through any point B draw B E parallel to A P, and upon B E set
off the distance B C equal to B A ; then join A C^ and it will bisect
the angle BAP.
Again, set off, upon B E, from C, C D = C A ; join A D, and it
will bisect C A P, or quadrisect BAP.
Again, set off, upon B £, D E = D A ; join £ A ; and E A P will
be ^ of B A P : and so on. (Fig. 70.)
Prob. Y. At a given point A in a given line A B, to make an angle
equal to a given angle C.
From the centres A and C, with the same radius, describe the arcs
D E, F G, Then, with radius D £, and centre F, describe an arc, cut
ting F G in O. Through G draw the line A G ; and it will form the
angle required, (Fig. 71 .)
Prob. VI. To divide any given angle ABC into three equal parte.
From B, with any radius, describe the circle A C D A. Bisect the
angle A B C by B £, and produce A B to D. On the edge of a ruler
CHAP. III.] PRACTICAL GEOMETRY. 113
mark off the length of the radius A B. Lay the ruler on D, and
move it till one of the marks on the edge intersects B £, and the
other the arc A C in O. Set off the distance C G from G to F : and
draw the lines B F, and B O, they will trisect the angle ABC.
(Fig. 72.)
Prob. VII. To divide a given line A B into any proposed number
of equal parU,
Ist Method. Draw any other line AC, forming any angle with the
given line A B ; on which set off as many of any equal parts, A D,
D E, E F, F C, as the line A B is to he divided into. Join B C ;
parallel to which draw the other lines F G, EH, D I : then these
will divide AB in the manner required. (Fig. 73.)
2nd Method^ without drawing parallel lines. Let A B he the line
which is to be divided into n equal parts. Through one extremity
A draw any right line A D, upon which set off n f 1 equal parts, the
point D being at the termination of the (n + l)th part. Join DB
and produce it until the prolongation BE = B D. Let F be the
termination of the (n — l)th part. Join FE, and the right line of
junction will cut the given line AB in the point P, such that
B P =  A B ; and of course n distances each equal to B P set off
upon B A, will divide it, as required*. (Fig. 74.)
Pbob. VIII. To cut off from a given line A B, supposed to he
tery shorty any proportional part.
Suppose, for example, it were required to find the y^, ^^^ j\r, &c.
of the line A B, fig. 75. From the ends A and B draw AD, B C,
perpendicular to A B, and divide A D into twelve equal parts ; then
through these divisions 1, 2, 3, &c., draw lines \f 2^, &c., parallel
to A B. Draw the diagonal A C, and 1 d will be the ^^ of A B ;
2 c, ^j, and so on. The same method is applicable to any other
part of a given line.
Prob. IX. To make a diagonal scale^ say^ offeety inches, and
tenths of an inch.
Draw an indefinite line A B, on which set off from A to B the
given length for one foot, any required number of times ; and from
these divisions A, C, H, B, draw AD, CE, &c., perpendicular to AB.
On A D and B F set off any length ten times, and through these
divisions draw lines parallel to A B ; then divide A C and D E into
twelve equal parts, each of which will be one inch. Draw the lines
A 1, ^2, &c., and they will form the scale required; viz., each of
the larger divisions from E to G, G to F, &c., will represent a foot ;
each of the twelve divisions between D and £, an inch ; and the
* The truth of thie method is easily demonstrated. Through i the inter
mediate point of division, on a n, between f and d, draw i b. Then, because
D B « B K, and D I B 1 F, I B is parallel to F P. Consequently, b P : B a : :
I P : I A : : 1 : n, by construction.
I
114 PRACTICAL OBOMBTBY. (^r^^. .
several horizontal lines parallel to R C in the triangle £ C R, will be
equal to j\^, f%, ,»ff, &c., of an inch. (Fig. 76.)
Note. — If the scale be meant to represent feet, or any other onit,
and tenths and hiindredfks^ then D E must be divided into ten instead
of twelve equal parts.
Prob. X. To fitid the centre of a circle.
Draw any chord A B, and bisect it perpendicularly with the line
R D. Then bisect R D in C, which will be the centre required.
(Fig. 29.)
Prob. XI. To divide a given cirde into any number ofeoneentrk
parts^ equal to each other in area.
Draw the radius A B, and on it describe the semicircle Afe dB.
Divide AB into the proposed number of equal parts, 1, 2^ 3, &c.,
and erect the perpendiculars 1 rf, 2e, Sf &c., meeting the semicircle
in rf, e,yj &c. Then from the centre B, and ^nth radii Be/, Be, &c.
describe circles ; so shall the given circle be divided into the proposed
number of equal concentric parts. (Fig. 77.)
Prob. XII. To divide a given circle into any number qf partij
equal both in area and perimeter.
Divide the diameter Q R into the proposed number of equal parts
at the points S, T, V, &c. ; then, on one side of the diameter
describe semicircles on the diameters QS, QT, QV, and on the
other side of it describe semicircles on RV, RT, RS; so shall the
parts 17, 35, 53, 71, be all equal> both in area and perimeter.
(Fig. 78.)
Prob. XIII. To describe the circumference of a circle through three
given points^ A, B, C.
From the middle point B draw chords B A, B C, to the two other
points, and bisect these chords perpendicularly by lines meeting in O,
which will be the centre. Then from the centre O, at the distance
of any of the points, as O A, describe a circle, and it will pass
through the two other points B, C, as required. (Fig. 79.)
The same method may be employed for finding the centre of
a circular arc, by taking any three points in the same, as A, B,
and C.
Prob. XIV. To describe mechanically the circumference of a cireU
through three given points^ A, B,C, tchen the centre is inaccessible^ or the
circle too large to be described with compasses.
Place two rulers M N, R S, cross ways, touching the three points
ABC. Fix them in V by a pin, and by a tmasverse piece T. Hold
a pencil in A, and describe the arc B A C, by moving the angle RAN
so as to keep the outside edges of the rulers against the pins B C.
Remove the instrument R V N, and on the arc described mark two
CHAP. III.] PRACTICAL GEOMSTBY. lid
points, D, E, 80 that their distance Bhall be equal to the length B C.
Apply the edges of the instrument against D £, and with a pencil in
6 describe the arc B C, which will complete the circumference of the
circle required. (Fig. 80.)
Otherwise, — Let an axle of 12 or 15 inches long carry two unequal
wheels A and B, of which one, A, shall be fixed, while the other, B,
shall be susceptible of motion along the axle, and being placed at any
assigned distance, A B, upon the paper or plane on which the circle is
to be described. Then will A and B be analogous to the ends of a
conic frustum, the vertex of the complete cone being the centre (O)
of the circle (C D E F) which will be described by the rim, or edge, of
the wheel A, as it rolls upon the proposed plane. Then it will be,
as the diameter of the wheel A is to the difference of the diameters
of A and B, so is the radius of the circle proposed to be described by
A, to the distance, A B, at which the two wheels must be asunder,
measured upon the plane on which the circle is to be described.
The wheel B will evidently describe, simultaneously, another circle
(O H I K) whose radius will be less than that of the former bv A B.
(Fig. 81.)
Pbob. XV. On a ^ven chord ABto describe mechanicaUy an arc
of a large circle that shall contain any number of degrees.
Place two rulers, forming an angle A C B, equal to the supplement
of half the given number of degrees, and fix them in C* Place two
pins at the extremities of the given chord, and hold a pencil in C ;
then move the edges of this instrument against the pins, and the
pencil will describe the arc required. (Fig. 82.)
Suppose it is required to describe an arc of 50 degrees on the given
chord A B ; subtract 25 degrees (which is half the given angle) from
180, and the difference, 155 degrees, will be the supplement. Then
form an angle A C B of 155 degrees with the two rulers, and proceed
«s has been shown above.
Pbob. XVI. To describe the segment of a circle of large radius^ of
which the chord A B and versine C D are given.
Through D draw H I parallel to A B ; then join A D and D B, and
draw A H perpendicular to A D and B I perpendicular to B D, also
draw A d and B d perpendicular to A B. Then divide A C, B C, H D,
and I D, into any number of equal parts, 1, 2, 8, &c., and draw lines
joining the corresponding numoers in H I and A B ; also divide A c/,
B d, into the same number of equal parts, a, by c^ &c., and draw lines
from the point D to these last divisions, then will the points where
the lines a D, 6 D, c D, &c. cut the lines 1 1, 22, 33, &c., be so many
points in the required segment. (Fig. 83.)
Pbob. XVII. To find the length of any given arc of a circle^ A B.
From A and B, as centres, widi radius equal to a quarter of the
diord of the arc A B, describe arcs cutting the given arc in C, and its
diord ID D ; join C D, which will equal h^f the length of the aro A B
nearly. (Fig. 84.)
I 2
116 PRACTICAL GEOMETRY. [PART I.
Prob. XVIII. To draw a straight line equal in length to any given
portion of the circumference of a circle.
Let A B C D be the circle, and let A a, a ^, 6 B, be the portions of
the circumference of wliich the length is required ; draw^ the two
diameters B D and A C at right angles to each other, and at the end
of the latter draw the tangent line Ae ; then divide the radius EC
into four equal parts, and set off three of them from C to F on the
diameter produced ; then draw lines from F through the points a,d,B,
cutting the tangent line A^, in c, rf, e; so shall the lines Ac, c</, de,
be equal in length to the arcs A a, a 6, 3 B, and the whole line Ae
equal to the quadrant A B. (Fig. 85.)
Prob. XIX. To bisect any given triangle ABC.
Upon any one of the sides, as A B, describe a semicircle, which
bisect in the point D ; then from B as a centre with radius B D de
scribe an arc cutting A B in E, through which point draw the line
E F parallel with the side A C ; then will the line £ F bisect the
given triangle ABC. (Fig. 86.)
Prob. XX. To reduce a given rectilinear figure, ABCDEFOA,
of any number of aides, to a triangle of equal area.
Join any two alternate angles, as A C, and through the interme
diate angle B, draw B H parallel to A C, cutting one of the adjoining
sides in H, and join C H, then will the triangle C B a, added to the
figure, be equal to the triangle a A H taken out of the same ; in like
manner join H D, and draw G I parallel to the same through the in
termediate point O, producing it to cut the side A G (also produced)
in I, and join I D. Next join D F, and through E draw a line paral
lel thereto, cutting the side G F produced in K, and join D K ; lastly,
join D G, and draw parallel to the same through the point K a line
cutting the side A G produced in L, then join D L, and the tri
angle I D L will be equal in area to the given rectilinear figure
ABCDEFGA. (Fig. 87.)
Prob. XXI. To form a rectangle of the largest area, in a given
triangle, ABC.
Bisect any two sides A B and B C in D and E, and from those
points draw lines D F and £ G perpendicular to the third side A C,
and join D E, then will the rectanele D E F G be the largest which
can be inscribed in the given triangle. (Fig. 88.)
Prob. XXII. To form a square equal in area to a given triangley
ABC.
On the longest side produced set off C D, equal half the perpendi
cular height B E ; and at C erect the perpendicular C F ; then on
A D describe a semicircle cutting C F in G, and on Q G form the
square CGHI, which will be equal in area to the given triangle.
(Fig. 89.)
CHAP. III.] PRACTICAL OEOIIBTRY. 117
Pbob. XXIII. To find the 9ide of a square eqtuU in area to a given
rectangle^ A B C D.
Produce the lesser side A B of the rectangle till A E equal the
longer side AD; then describe upon A £ a semicircle cutting B C in
F, and join A F, which will be the side of the square required.
(Fig. 90.)
Pbob. XXIV. To find the side of a square egtmlto the difference
beiveen two given squares.
Let the sides of the two given squares be A and B. Then draw
the line C D equal to the lesser line B, and at one of its extremities
erect the indefinite perpendicular D £ ; then with radius equal
to A, and from C as a centre, describe an arc cutting this perpen
dicular in E ; then D E will equal the side of the square required.
(Fig. 91.)
Pbob. XXV. To find the side of a square equal in area to any
number of given squares.
Let the lines A, B and C be the sides of the given squares. Draw
DE equal to A, and at the end of it erect the perpendicular E F equal
to B ; join D F and perpendicular to it, from the point F, erect the
perpendicular O F equal to C ; then join D O, which will be equal
to the side of a square equal in area to the three squares on A, B,
and C. (Fig. 92.)
Pbob. XXVI. To find the side of a square nearly equal in area to
a given circle, A B C D.
Draw the two diameters, AC, B D, at right angles to each other ;
then bisect the radius E C in F, and through D and F produce the
straight line D G cutting the circle in G, then will D G equal the side
of the square required. (Fig. 93.)
Pbob. XXVII. Given the side of a regular polygon of any number
of tides (not exceeding twelve% to find the radius of the circle in which
it may be inscribed.
Multiply the given side of the polygon by the number which stands
in column 6, opposite its proper name in the annexed Table of
Polygons ; the product will be the radius required.
Thus, suppose the polygon was to be an octagon, and each side 1 2,
then 13065628 X 12 = 156687536 would be the radius sought.
Take 15*67 as a radius from a diagonal scale, describe a circle, and
from the same scale, taking off 12, it may be applied as the side of
an octagon in that circle.
Pbob. XXVIII. Given the radius of a circle, to find the side of any
regular polygon (not having more than twelve sides) inscribed in it.
Multiply the given radius by the number in column c, standing
opposite the number of sides of the proposed polygon : the product
is the length of the side required.
Thus, suppose the radius of the circle to be 5, then 5 x 1*732051
as 8*66025, will be the side of the inscribed equilateral triangle.
118
PKAOTICAL OSOMITBT.
TABLE OP POLYGONa
[Fi
6 "5
Multipliers
Radius of
Fac
Names.
for areas.
circmD. cir.
fori
Z,'^
(«)
(ft)
(
8
Trigon
04330127
05773503
173S
4
Tetragon, or Square
10000000
07071068
141^
5
Pentagon
17204774
08506508
117^
6
Hexagon
25980762
10000000
100(
7
Heptagon
36339124'
1 1523824
086'
8
Octagon
48284271
13065628
076,
9
Nonagon
61818242
14619022
068
10
Decagon
76942088
16180340
061J
11
Undecagon
98656399
17747324
056;
12
Dodecagon
111961524
19318517
051'
Pros. XXIX. To reduce a simple rectilinear figure to a i
one upon either a smaller or a larger scale.
Pitdi upon a point P any where about the given figure A B
either within it, or without it, or in one side or angle; but m
middle is best. From that point P draw lines through all the i
upon one of which take P a to P A in the proposed proportion
scales, or linear dimensions ; then draw a b parallel to A B,
B C, &c. ; so shall abode be the reduced figure sought,
greater or smaller than the original. (Hutfons Mens.) (Fig. S
Otherwise to Reduce a Figure by a Scale, — Measure all the
and diagonals of the figure, as A B C D E, by a scale ; and lay
the same measures respectively from another scale, in the pro]
required.
To Reduce a Map^ Design^ or Figure^ by Squares. — Divide tl
ginal into a number of little squares, and divide a fresh paper,
dimensions required, into the same number of other squares,
greater or smaller, in the proportion required. Then, in every
of the second figure, draw what is found in the corresponding
of the first or original figure.
The cross lines forming these squares may be drawn with a
and rubbed out again after the work is finished. But a more
and convenient way, especially when such reductions are
wanted, would be to keep always at hand frames of squares
made, of several sizes ; for, by only just laying them down up
papers, the corresponding parts may be readily copied. These :
may be made of four stifiT or inflexible bars, strung across with
hairs, or fine catgut.
When figures are rather complex, the reduction to a differen
will be best accomplished by means of such an instrument as I
sor Wallace's Eidograph^ or by means of a Pantograph^ an i
ment which is now considerably improved by simply changic
place of the fulcrum. See the Mechanics' Orade^ Part II. p. ;
CfliP.IV.J WS10HT8 AND IfEASURKS. 119
CHAP. IV.
MENSURATION.
Sect. I. Weights and Measures,
MmuEATiON is the application of arithmetic to geometry, by
wliicb m ire enabled to discover the magnitude and dimeneious of
tnj geometrical figures, whether solid or superficial. To enable us
to expitis this magnitude in determinate terms, it is necessary to
womeflome magnitude of the same kind as the unit, and tlien, by
itatiDg how many times the given magnitude contains that unit, we
obtain its measure.
The different species of magnitude which have most frequently to
be determined, are distinguishable into six kinds, viz. : —
1. Length.
2. Surface.
3. Solidity, or Capacity.
*. force of Gravity, commonly called Weight.
5. Angles.
6. Time.
The several units assumed as the standards of measurement of each
of then particular species of magnitude, are entirely arbitrary, and,
ttoie^ently, vary among different nations. In this kingdom they
w heea fixed by Act of Parliament*, and are as follows, viz. : —
Length is a yard.
. Surface is a square yard, the ^g^^^ th of an acre.
I (Solidity is a cubic yard.
Tie standard of < i^^P???^ '' "" ^^l'"'''
mmuuMuix "» \ height IS a pound.
I Angular measurement is a degree, the d60th part
of the circumference of the circle.
Time is a day.
ne values of the whole of the above are determined directly or
Wifectly by comparison with the length of a pendulum, which in
^ latitude of London, placed at the level of the sea, and in a
* 5 Gso. IV. c. 74 ; which took effect Itt January, 1826.
120 WBIOHTS AND 1IBA8URBS. [r^^.
yacuuTD, would vibrate seconds of mean time. The length of such
a pendulum being, to the length of the standard yard, as 39*1393
inches are to 36.
Since the passing of this act, however, some very elaborate and
scientific experimenis of Mr. Francis Daily have shown that errors
of sufficient moment to be taken into the account in an inquiry of
this kind, render the above proportion inaccurate*. We do not, in
fact, yet know the length of a seconds* pendulum at London, vibrat
ing in the circumstances proposed.
The following standard yards, made with great accuracy, give the
annexed results : —
Inches.
General Lambton's scale, used in India ... 35*99934
Sir George Shuckburgh's scale 35*99998
General Ray's scale 36*00088
Royal Society's standard 3600135
Ramsden'sbar 36002*9
Its copy, at Marischal College, Aberdeen... 36*00244
1. MEASURES OF LENGTH.
Inchcft. Feet.
12 = 1 Yard..
o/» o __ I Rod*
<5t> — d — 1 orPoie*.
198 = 16J = 5J == 1 Furicmg..
7920 = 660 = 220 = 40 = 1 Mile.
63360 = 5280 = 1760 = 320 = 8 = 1
The mean length of a degree of latitude measured on the terrestrial
meridian, is 69*0444 imperial miles; the 60th part of which, or
6075*6 feet, is the length of a nautical or geographical mile, three of
which are equal to a league. And the length of a degree of lo/igi
tude, measured upon the equator, is 69*1555 imperial miles.
An inch is the smallest lineal measure to which a name is nven ;
but subdivisions are used for many purposes. A mong mechanics the
inch is commonly divided into eighths. By the officers of the revenue,
and by scientific persons, it is divided into tenths.^ hundredthn^ &c.
Formerly it was made to consist of 12 parts, called lines^ but these
have properly fallen into disuse.
Particular Measures of Length.
A Nail = 2\ Inches \
Y A — • 4 O f i ^^^ ^^^ measuring cloth of all kinds
Ell = 5 Quarters )
Hand = 4 Inches Used for the height of horses.
Fathom = 6 Feet Used in measuring depths.
Link = 7 Inc., 92 \ Used in Land Measure, to faciliti
hdths. } computation of the content,
Chain =100 Links ) square chains being equal to an ac
* See footnote, p. 223.
f
CRIP. IV.J WBIOHTS AND MKA8UBES. 121
2. MEASURES OF SURFACES.
SqoRliKfaa.
144 =
1296 =
39204 =
1568160 =
Sq.Feet.
1
9 =
272i =
10890 =
Sq. Yards,
1 orRodi.
30 = 1 Rood..
J210 = 40= 1 a™.
6272640 =
43560 =
4840 = 160= 4= 1 mL
4014489600 = S
^878400 =
3097600 =102400 = 2560 = 640 = 1
d. HBASUBE8 OF SOLIDITY AND CAPACITY.
Division I. Measures 0/ Solidity,
Cubic Inchet. Cubic Feet.
1728 = 1 Cubic Yard.
46656 = 27 = 1
Division II,
Mntures of Capacity for aU liquids^ and for aU dry goods^ except
wc^oiare comprised in the third division,
Tbe imperial gallon (the standard for all measures of capacity)
contains 10 pounds imperial avoirdupois weight of distilled water,
'wghedin air at 62^ Fahr. (the barometer being at 30 inches) ; con
'wiuently, its capacity is 277*274 imperial cubic inches.
4 Gills = 1 Pint = 34659 Cubic Inches.
2Rnt8 = 1 Quart = 69 318 „ „
4 Quarts = 1 Gallon = 277*274 „ „
2 Gall. =1 Peck = 554548 „ „
8 Gall. = 1 Bushel = 2218192 „ „
8 Bush. = 1 Quarter = 10269 Cubic Feet.
5Qr8. = 1 Load = 51*345 „ „
The four last denominations are used for dry goods only. For
liquids leveral denominations have been heretofore adopted, viz. : —
^or beer, the firkin of 9 gallons, the kilderkin of 18, the barrel
0(36, the hogshead of 54, and the butt of 108 gallons. These
will probably con tin ae to be used in practice. For wine and
"pvits, there are, the anker, runlet, tierce, hogshead, puncheon,
PpC) butt, and tun; but these may be considered rather as the
i^Anies of the casks in which such commodities are imported, than
•• expressing any definite number of gallons. It is the practice to
8>oge all such vessels, and to charge them according to their actual
cwitcnt.
Flour is sold, nominally, by measure, but actually by weight,
'^Itoned at 7 lb. aToirdupois to a gallon.
122
WBI0HT8 AND IfEASURIS.
[PABT I,
Division HI.
Imperial Measures of Capacity for coalsy 'culm^ lime, fish^ potaioet^
fruity and other goods, commonly sold by hbapbd measukb :
2 Gallons = 1 Peck
8 Oallons = 1 Bushel
3 Bushels = 1 Sock
12 Sacks == 1 Chald.
28154 I ^^^^^ Inches, nearly,
gt I Cubic Feet, nearly.
The goods are to be heaped up in the form of a cone, to a height
above the rim of the measure of at least f of its depth. The out
side diameter of measures used for heaped goods are to be at least
double the depth ; consequently not less than the following dimen
19i inches.
Bushel
Halfbushel 15 inches.
!
Peck 12 J inches.
Gallon 9 inches.
Halfgallon 7f Inches.
The imperial measures described in the second and third divisiont
were established by Act 5 Geo. IV. c. 74. Before that time there
were four different measures of capacity used in England, yiz. : —
2. For malt liquors, the gallon of which con ) ooo
tained j ^^^ "
3. For com and all other dry goods, not  aoo.o
heaped, the gallon of which contained ... ) " "
4. For coals, which did not differ sensibly from the imperial
measure.
Hence, with respect to wine, ale, and com, it will be expedient
to possess a
Table of Factors,
For converting old measures into new, and the contrary.
By Decimals.
By Vulgar Fractions
nearly.
Com
Measure.
Wine
Measure.
Ale
Measure.
Corn
Mea
sure.
Wine
Mea
sure.
Ale
Mea
sure.
To convert old ) .96943
measures to new. S
83311
101704
n
f
n
To convert new J ^ 3^ 3
measures to old. S
120032
98324
i\
i
^
N.B. For reducing the prices, these numbers must all be revenad.
GHAP. IV.] WUOHTS AND IfSAtUBBS. 123
4. MBA8URE8 OF WEIGHT.
Division L Avoirdupois Weight.
The standard of weight is the avoirdupois pound, the value of
'vhich is determined by its heing the weight of 27'7274 cuhic inches
^f distilled water, weighed in air at 62"^ Fahr., with the harometer at
SO inches.
Avoirdupois weight is used in almost all commercial transactions,
^md in the common dealings of life; its divisions are as follows : —
Dndum.
Ounees.
16
s:
1
Pounds.
256
S5
16
=c
1
QiuDten.
7168
=
448
=s
28
1
Hundred
weighu.
28672
=
1792
=
112
=
4
5= 1
Ton.
573440
z^
35840
=
2240
=
80
= 20
Particular weighu belonging to this Division.
cwt. qr.
lb.
8 Pounds =
1 Stone
z=
8
Used for meat.
1* .>
=
1 «
s=
14 \
2 Stone
s=
1 Tod
s=
1
eTod
=
1 Wev
=
1 2
14
I)
Used
in the wool trade.
2 Wejs
=s
1 Sack
==
3 1
12 Sacks
2=
1 Last
=5
30
Division IL Troy Weight,
For weighing gold, silver, and precious stones (except diamonds),
troj weight is employed, its divisions are —
Penny
OtaiBi (gn.) i*«ighU (dwts.)
24
=s
1
OUBCM(OI.)
480
5=
20
= 1 Pound (lb.)
5760
=
240
= 12 = 1
But troy weight is also used hv apothecaries in compounding medi
cines; and 18 then divided as follows, viz. : —
Onant(gr.) Seruplo (3.)
20 = 1 Drachms
(3.)
60
480
5760
i
3 =
24 =
288 ^
1
8
96
Ounces (3 )
= 1
= 12
Pound (lb.)
= 1
The troy pound is equal to the weight of 22*8157 cuhic inches of
distilled water, weighed in air at 62^ Fahr., and the harometer at
30 inches; and is, therefore, less than the avoirdupois pound, in the
proportion of 144 to 175.
124 WEIGHTS AND MEASURES. [PABT I.
OS. dwts. gn.
1 lb. avoirdupois = 14 11 15i troy = 7000 troy grains.
1 oz. „ == 18 5$ „ = 4375 „
1 dr. „ =01 3 „ = 27343 „
1 trov lb. = 0822857 avoirdupois lb.
1 avoir, lb. = 1215271 troy lb.
For scientific purposes the grain only is used ; and sets of weights
are constructed in decimal progression, from 10,000 grs. downwards
to i^J^th of a grain.
The carat^ used for weighing diamonds, is 3^ grains. The term,
however, when used to express the fineness of gold, has a relative
meaning only. Every mass of alloyed gold is supposed to be divided
into 24 equal parts; thus the standard for coin is 22 carats fine, that
is, it consists of 22 parts of pure gold, and 2 parts of alloy. What
is called the new standard^ used for watchcases, &c. is 18 carats
fine.
5. Angular Measure; or. Divisions of the Circle.
60 Seconds " =1 Minute, denoted by '
60 Minutes = 1 Degree, „ °
30 Degrees = 1 Sign „ '
90 Degrees = 1 Quadrant.
360 Degrees, or 12 Signs = 1 Circumference.
Formerly the subdivisions were carried on by sixties ; thus, the se
cond was divided into 60 thirds, the third into 60 fourths, &c. At
present, the second is more generally divided decimally into lOths,
lOOths, &c. The degree is frequently so divided.
6. Measure of Time.
60 Seconds = 1 Minute.
60 Minutes = 1 Hour.
24 Hours = 1 Mean Solar Day.
23 H. 56 M. 35 S. = 1 Sidereal Day.
7 Days = 1 Week.
28 Days = 1 Lunar Month.
28, 29, 30, or 31 Days = 1 Calendar Month.
12 Calendar Months = 1 Year.
365 Days = 1 Common Year.
366 Days = 1 Leap Year.
365jDavs = 1 Julian Year.
365 D. 5 H. 48 M. 45 S. = 1 Solar Year.
365 D. 6 H. 9 M. 11 S. =1 Sidereal Year.
A solar day is the time that elapses between two successive transits
of the sun over the same meridian, and is not always of equal dura
tion, being longer at some seasons of the year than at others ; the
difference between the actual length of a solar day and the mean
length, is called the equation of time. A sidereal day is the interval
of time that elapses between two successive transits of any fixed star
CHiP, IV.]
WEIGHTS AND MRA8UBBS.
125
OTer the same meridian, and is tbe most uniform of all astronomical
pen'ods, neither theory nor observation having detected the slightest
TSjiatioD in its length. A solar year is the time in which the earth
passes through the twelve signs of the zodiac, and is the natural year,
Dccause it always keeps the same seasons in the same months. The
wd^eal year is the time that elapses between the eartb leaving any
fised star and returning to it again.
In 400 years, 97 are leap years, and 303 common.
Tbe same remark, as in the case of angular measure, applies to tbe
mode of subdividing the second of time.
COMPARISON OF ENGLISH AND FRENCH WEIGHTS AND MEASURES.
The following is a comparative Table of the Weights and Measures
^i England and France, which were published by tbe Royal and
dJoitral Society of Agriculture of Paris, in the Annuary for 1829,
^uid founded on a Report, made by Mr. Mathieu, to the Royal Aca
demy of Sciences of France, on the bill passed the 17th of May, 1824,
^■^latiTe to the Weights and Measures termed " Imperial," which are
xaow used m Ghreat Britain.
ENGLISH.
1 Inch (l36th of a yard)
1 Foot (l3rd of a yard
'Yard imperial
Fathom (2 yards) .
^ole, or perch (5^ yards
Fnrlong (220 yards)
Mile (1760 yards)
FRENCH.
1 Mill;
1 Centimetre
1 Decimetre .
IMetre .
^yriamctre ,
Measures of Length.
FRENCH.
2*539954 centimetres.
30479449 decimetres.
091438348 metre.
182876696 metre,
502911 metres.
20116437 metres.
1609*3149 metres.
ENGLISH.
003937 inch.
0393708 inch.
3937079 inches.
3937079 incbes.
32808992 feet.
1093633 yard.
6*2138 miles.
Square Measures.
ENGLISH.
J Yard sqoare
Jod (square perch)
1 Bood (1210 yards square)
' Acre (4840 yards square)
FRENCH.
J Metre square
\^
IHeetare . .
FRENCH.
0836097 metre square.
25*291939 metres square.
10116775 ares.
0404671 hectare.
ENGLISH.
1196033 yard square.
098845 rood.
2473614 acres.
126
WnOHTS AND ICEABOBKt.
[pAWr I
Solid Measures,
ENGLISH.
1 Pint (l8th of a gallon)
1 Quart (l4th of a gallon)
1 Gallon imperial .
1 Peck (2 gallons)
1 Bushel (8 gallons)
1 Sack (3 bushels)
1 Quarter (8 bushels)
1 Chaldron (12 sacks)
FRENCH.
1 Litre . . = .
I Decalitre . . . . = .
1 Hectolitre . . . = ,
Weights.
ENbl.ISH TROY.
1 Grain (l24th of a pennyweight) =
1 Pennyweight (1 20th of an ounce) =
1 Ounce (11 2th of a pound troy) =:
1 Pound troy imperial . =
ENGLISH AVOIRDUPOIS.
1 Drachm (llOth of an ounce) . =
1 Ounce (l16th of a pound) =
1 Pound avoirdupois imperial =
1 Hundred weight (112 pounds) =
1 Ton (20 hundredweight) . =
FRENCH.
FRENCH.
0567932 litre.
M35864 litre.
454345794 litres.
908G9I59 litres.
36347CG4 litres,
109043 hectolitre.
2907813 hectolitres.
1308516 hectolitres.
ENGLISH.
1760773 pint
02200967 gallon.
22009667 gallons.
22009667 gallons.
FRENCH.
0*06477 gramme.
1*55456 gramme.
. 310913 grammes.
03730959 kilogramme.
FRENCH.
17712 gramme.
. 283384 grammes.
04534148 kilogramm*.
. 5078246 kilogiammm.
1015649 kilogrammes.
ENGLISH.
1 Gramme
Kilogramme
15438 grains troy.
0643 pennyweight.
0032 16 ounce troy.
268027 pounds troy.
2*20548 pounds avoirdupoil
Angular Measure,
In France, the centesimal division of the circle is frequently em
ployed, in which the whole circumference is divided into 400 degre«
each degree into 100 minutes, each minute into 100 seconds, &c.
CENTESIMAL. ENGLISH.
1 Degree . . . , = . 54 minutes.
1 Minute . . . = . 32*4 seconds.
1 Second . . . . = . *324 second.
ENGLISH.
1 Degree
1 Minute
1 Second
CBNTESIMAL.
1^ degree.
185185 minute.
8*08641 seconds.
CHAP. IV.] MBOTUJEUTION OF SUPBRFICIE8. 12?
Sect. II, Mensuration of Superficies*
TRIANGLES.
Let by c, and dy represent the three sides of a triangle (see
fig 95); 0j y, and ^, the angles opposite those sides respectively;
and h the perpendicular height from the vertex to the hase b;
then the
area =  & A,
or = I ^ c . sin ^ = ^ c (/ . sin jS = ^ </& . sin 7.
Or, = a/ </^ — f — — H  ) • 2 where b is the greatest
side and c the least
If half the sum of the three sides or = », then the
area = V» {s — A) . (a — c) . (« — d)y and the
log of area =i{log« 4 log (« — i) f log(« — c)  log (« — «?)}.
For the method of obtaining the unknown sides or angles of
triangles from those which are known, see Chap. V., Sect. IV.
QUADRILATERALS, OR FOURSIDED FIGURES.
Square^ Rectangle^ Rhombus and Rhomboid: — To obtain the area,
naltiplj the perpendicular height (^, fig. 96) by the base {b) on
which it falls.
The area of a Trapezium is best found by dividing it into two
triangles (as fig. 17), the areas of which may be found by the
foregoing rules.
The area of a Trapezoid (fig. 16) may either be found in the same
way, or by multiplying half the sum of the two parallel sides (a f b)
by the perpendicular distance between them (Ji).
POLYGONS.
To obtain the area of any regular polygon, having less than twelve
sides, multiply the square of one of the sides by the number found
in colamn a, of the Table of Polygons, page 118. Or, generally, if
/ =: the length of one of the equal sides, and n the number of them;
then the
area = /^ • tan
4
^90 n— ISOX
The area of an irregular polygon may be determined by dividing
it into triangles, as in fig. 97. Or by forming one triangle equal in
area to it, by Prob. XX., page 116.
CIRCLES.
Let d represent the diameter, e the circumference, a the area,
mnd/» := 3*14159 (see Table V. in the Appendix) ; then the
128 IfBNSURATION OF 8UPBRFICIB8. [PART I.
_. , c 4a ^ /a
Diameter =:(/ =  = — =2 a/~«
p c ^ p
4a
Circumference = c =/? {/ = — = 2 ^p a.
Area = « ='^,^ = / = ^' = 'TSS*./'.
4 4/? 4
^ "" 5 "~ rf* "~ 4a'
Circular Arcs. — If r represents the radius, and d the diameter of
the circle; a the sine, and r, the versed sine of the arc; c the chord
of half the arc, v^ the versed sine of half the arc; and m the mea
sure in degrees of the whole arc ; then
the length of the arc = 0174533 rm;
°'=2rf^^^^^ nearly;
8c — 2«
or = nearly.
And the following relations between the several quantities, firom
which any one of them may be obtained, will be found useful, viz.: —
c, =rN/i^^=7 .' (1.)
c=n/ ^^H^^ (2.)
«= N/r'^(rt?J* (3.)
^=f + t^x (*.)
Circular Sectors. — Let d represent the diameter of the circle,
/ the length of the arc of the sector^ and m its measure in degrees;
then the
area =  r /.
or = 00218 J^w.
Circular Segments, —Let d represent the diameter of the cirdc,
© the versed sine, c, the chord of the whole arc, and c^ the chord of
half the arc; then the
area =  {^{dv — ©^j ^ ^ ^dv} nearly;
or = 1 1? (</«? — 1^ c^) nearly;
<>r =^a<^(<^i +T<^2) nearly;
or = JtJ N/(i<?i^ + iv^) nearly.
* Table VI. in the Appendix contains the diameter, circumference, area, and
length of the side of an equal square to circles from 1 to 100*76 in diameter.
CHIP. IV. j MENSUBATION OF 8UPBBFICIE8. 1 29
Or the area of any segment, as fig. 98, may be obtained by finding
(lie area of the sector A BCD, and subtracting from it tbe area of
the triangle A C D.
Tbe area of a circular zone^ as £ F G H, may be found by subtract
ing the area of the segment GH I from tbe area of tbe segment
E6IHF.
PABA30LA.
The area of a parabola is equal to twotbirds of the product of tbe
l«e X the perpendicular height.
For the relations between die abscissee and ordinates in tbe para
bola, and the method of deriving one from the other, see Chap. VI.
page 172.
Psrahdic Arcs, — Let x be the abscissa a 5, ^g, 99, measured from
the vertex a, and y the corresponding right ordinate cb; then the
length of the btc cadis
= ^y/(fri^) nearly;
= {V(/
h*^)^
4
y
Parabolic Frustunij or Zone, — The word zone is here used to
clcDotethe space ede^ (fig. 99), contained between the two parallel
doable ordinates cd and e^. Let ^j = the length of the double
^'fliMte ed^ and y^ = the length of e^y and d = the perpendicular
datance between them bf; then the
S ..3
area of the zone = 4 ^ ^^, ^^ •
ELLIPSE.
The area is equal to the product of the transverse diameter x the
wnjngate diameter x 785398.
'^periphery or circumference is equal to the sum of the trans
'eweand conjugate diameters x 1*57079.
The foregoing is only an approximation to the periphery, although
wfidently near the truth for ordinary purposes ; but where greater
■ewracy is required the following series may be employed. Let t
'^P'ttcnt the transverse axis, c the conjugate, p = 3* 14159, &c.,
«d <^= 1 — ; then
r
^j, d 8d' SK5d^ 3\5\7d*
win be the periphery.
EBipOe Segments, — To find tbe area of tbe elliptic segment abc
(6g. 100), find tbe area of the corresponding circular segment, dbe^
Meribcd on the same axis (bf) to which tbe cutting line or base of
K
130 IfBNSUBATION OF 80LID8. [PAB
the segment ae is perpendicular. Then, as this axis (hf) :
other axis (ph) :: the circular segment (dbe) : the elliptic
ment (a be).
Elliptic Arcs. — Let t represent the semitransverse, and e
semiconjugate diameters of the ellipse, and d the distance of
ordinate from the centre, then the length of the arc conta
hetwecn the ordinate and the parallel semiaxis will be
'^'''e?^'^iO^'''' 112?^ ^ + *^
f^ — c^
or make ^ — = r ; then the
length of the arc = €? ^ / —5 — . ^ nearly.
HYPERBOLA.
The area of an hyperbola or hyperbolic segment may be founc
follows : — let t and c represent the semitransverse and semicoi
gate diameters, 2y the double ordinate which cuts off the segmt
and a its abcissa ; also g = : then the
area = 2:»yfi 1 ?1 ^ &c}
^^^ 3.5 3.5.7 3.5.7.9 ^
or =£^(4 ^(2tx 4. f ar^) 4 ^Ztx] nearly.
Hyperbolic Arcs. — The notation being the same, the follow
approximation may be employed to obtain the length of an hy]
bolic arc.
120c^ + (19/* + 21c')4«
^'^ = 120c^/K9f^ + 21c*)4^ +y' ^«^^y
Sect. III. Mensuration of Solids.
PARALLELOPIPEDON, PRISM, OR CYLINDER.
Surface. — Multiply the perimeter of one end by the lenstl:
perpendicular height, to which add the area of the two en^
sum will be the surface.
Solidity = the area of the base x the perpendicular height
Pyramid or Cone.
Surface. — Multiply half the perimeter by the slant height, to w
add the area of the base, the sum is the surface.
Solidity = the area of the base x onethird the perpendic
height.
CBihlV.] IfBNSUaATION OF SOLIDS. 131
Pnutumofa Pyramid or Cone, Surface, — Multiply half the sum
of the perimeters of the two ends hy the perpendicular height.
SdidHy. — ^Add a diameter or side of the greater end to one of the
leas, ind from the square of the sum subtract the product of the said
two ditroeters or sides ; then multiply the remainder by onethird of
the height, and this product by '785398 for circles, or by the proper
Doitiplier for polygons ; the last product will be the capacity.
That is, let D equal the greater diameter, d the less, h the perpen
^Iw beight, and p = "785398 for cones, or for any pyramid the
proper multiplier from column (a) in the table at page 118, then
Solidity = jjoA(D* + Drf + rf«).
SPHBBB.
S^irface = the diameter x the circumference, = the square of
the diimeter x 3*14159, = the square of the circumference
X •3183.
Sdiditp = the cube of the diameter x '5236, = the cube of the
cbmifercnce x 01688.
8fkencd Segment. — Let d equal the diameter of the sphere, r
the ndius of the base of the segment, and h its height ; then the
otnednafaee = 3*14159 <f A.
Solidiiy = 05236 k^ (3 </ — 2 A) ;
or = 05236 A (3 r' j h").
The surface and solidity of a spherical zone may be obtained by
^ing the difference between the two segments.
CONOIDS.
A conoid is the solid generated by the revolution of a conic section
Jj^nt one of its axes, and is called a epheroidy paraboloid^ or hyper
^*fc*rf, according to the section from which it is produced.
. Spheroids, — When the ellipse revolves about its transverse axis, it
*• called an oblong or prolate spheroid; when about its conjugate axis,
^'^ (Hate spheroid; and when about any other of its diameters, a
^^ivemd spheroi^ij in which latter case its figure is somewhat re
^^ttbling a heart. To obtain the solidity of a spheroid, multiply the
?^I^ttre of the revolving axis by the fixed axis, and the product
'^ 05236.
Pmdfoloid. — Let y be the radius of the circular base, and x the
^titode of the solid ; then
Surface = ^i^^ {(/ + 4:r»)?  y^}.
Solidity = V5708y^x.
y^HptrboUnd, — Let t equal the transverse axis, r the radius of the
^*^ « the altitude, and p the parameter ; then the
Solidity = J /^^ • . ^
K 2
132
MENSURATION OF SOLIDS.
[PARr
THE REGULAR OR PLATONIC SOLIDS.
The regular or Platonic bodies are five in number, and have all
been described at page 111.
1. To find either the surface or the solid content of any of M
regular bodies. — Multiply the proper tabular area or surface (talf:^
from column (a) in the following table) by the square of the lin ^
edge of the solid, for the superficies. And
Multiply the tabular solidity in column {h) of the table by the cvsl
of the linear edge for the solid content.
Surfaces and Solidities of Regular Bodies^ the side being unity or 1
No. of
sides.
Name.
Surface.
Solidity.
(*)
4
6
8
12
20
Tetraedron
Hexaedron
Octaedron
Dodecaedron
Icosaedron
17320508
60000000
34641016
206457288
86602540
01178513
10000000
04714045
76631189
21816950
2. The diameter of a sphere being given^ to find the side of 4^
of the Platonic bodies^ that may be either inscribed in the sphere '9
circumscribed about the sphere^ or that is equal to the sphere, — ^^
tiply the given diameter of the sphere by the proper or correspond*
number, in the following table, answering to the thing sought, ^^
the product will be the side of the Platonic body required.
^
The diam. of a
sphere being 1 ;
the side of a
Tetraedron
Hexaedron
Octaedron
Dodecaedron
Icosaedron
That may be
inscribed in the
sphere, is
That may be cir
cumscribed about
the sphere, is
08164966
05773503
07071068
03568221
05257309
24494897
10000000
12247447
04490279
06615845
That it equal
to the sphere.
164394A0
08059958
10356300
04088190
06214433
3. The side of any of the five Platonic bodies being given^ to find
the diameter of a sphere^ that may either be inscribed in thai body^ or
circumscribed about it^ or that is equal to it, — As the respective
number in the table above, under the title inscribed^ eircumscribedy
or equcdy is to 1, so is the side of the given Platonic body to the
diameter of its inscribed, circumscribed, or equal sphere.
4. The side of any one of the fioe Platonic bodies being given^ to
find the side of the other four bodies, that may be equal in solidi^ to
that of the gioen body. — As the number under the title equal in the
:aiP. IV.] MENSURATION OP SOLIDS. 133
mMl colomn of the table above, against the given Platonic body, is to
be number under tbe same title, against the body whose side is
^agfat, 80 is tbe side of the given Platonic body to the side of the
M)dj sought.
Besides these tbere are thirteen demiregular bodies, called Solids
^Archimedes, Tbey are described in the Supplement to Lidonne's
f^M$ de tons les Diviseurs des Nombres^ &c., Paris, 1808 ; twelve
»f them were described by Abraham Sharp, in his Treatise on
E^olycdra.
re nND THE CONTENTS OF SURFACES AND SOLIDS NOT REDUCIBLE TO
▲BY KNOWN FIGURE, BY THE EQUIDISTANT ORDINATE METHOD.
The general rule is included in this proposition, viz. : — If any
ri^btline be divided into any even number of equal parts, 1, 2, 3, 4,
&C., (fig. 101,) and at the points of division be erected perpendicular
oxdiDates 1 A, 2 B, 3 C, &c., terminated by any curve A C G : then,
iC a be put for the sum of the first and last ordinates, 1 A, 7 O, e for
die sum of the even ordinates, 2 B, 4 D, 6 F, &c., viz., the second,
fourth, sixth, &c., and o for the sum of all the rest, 3 C, 5 E, &c.,
"VIZ., the third, fifth, &c., or the odd ordinates, excepting the first and
la«t: then, the common distance 12, 2 3, &c., of the ordinates
l>cing multiplied into the sum arising from the addition of «, four
^TDes e, and twice o, OTte third of the product will be the area 1 A G 7,
▼wy nearly.
_ / # + 4^ h 2o
That 18, —  — ^ . D = area, D being = A C = CE, &c.
3
The same theorem will equally serve for the contents of all solids,
by oaog the sections perpendicular to the axis instead of the ordi
iistet. The proposition is quite accurate, for all parabolic and right
^ areas, as well as for all solids generated by the revolutions of
^ic sections or right lines about axes, and for pyramids and their
^Wims. For other areas and solidities it is an excellent ap
proximation.
The greater the number of ordinates, or of sections, that are taken,
^ more accurately will the area or the capacity be determined.
Bot b a great majority of cases^t^e equidistant ordinates, or sections,
^ lead to a very accurate result.
134 PLANE TRIOONOMBTRY. [PAKT I.
CHAP. V.
TBiaONOMETRY.
Sect. I. Definitions and lVi</onotnetriciU Formulae.
1. Plane Trigonometry is that branch of mathematics by which
we learn how to determine or compute the unknown parts of a
plane, or rectilinear triangle, from those which are known, when that
18 possible.
Every triangle consists of six parts, viz , three sides, and three
angles opposite those sides. And any three of these being given
(excepting only when the three angles are given) the others may
always be determined from them.
The determination of the mutual relation of the sincs^ tangents,
secants^ &c., of the sums, differences, multiples, &c., of arcs or angles;
or the investigation of the connected formulae, is also usually classed
under plane trigonometry.
2. Let ACB (fig. 102) be a rectilinear angle: if about C as a
centre, with any radius C A, a circle be described, intersecting C A,
CB, in A, B, the arc AB is called the measure of the angle ACB.
3. The circumference of a circle is supposed to be divided or to
be divisible into 360 equal parts, called degrees; each degree into GO
equal parts, called minutes; each of these into 60 equal parts, called
seconds; and so on to the minutest possible subdivisions. Of these,
the first is indicated by a small circle, the second by a single accent,
the third by a double accent, &c. Thus, 47° 18' 34" 45''', denotes
47 degrees, 18 minutes, 34 seconds, and 45 thirds. The number of
degrees, minutes, seconds, &c., contained in the arc AB of the circle
described from the angular point C, and which is contained between
the two legs AC, A B, is called the measure of the angle ACB,
which is then said to be an angle of so many degrees, minutes,
seconds, &c. Thus, since a quadrant, or quarter of a circle, contains
90 degrees, and a quadrantal arc is the measure of a right angle, a
right angle is said to be one of 90 degrees.
4. The complement of an arc is its difference from a quadrant,
as B E ; and the complement of an angle is its difference from a right
angle, as £ C B.
5. The supplement of an arc is its difference from a semicircle,
as A' £ B ; and the supplement of an angle is its difference from two
right angles, as A' C B.
CHIP, v.] PJLANB TBIOONOIIBTKY. 135
6. The tine of an arc is a perpendicular let fall from one extre
nitj opon a diameter passing through the other, as B D.
7. the versed eine or versine of an arc is that part of the dia
eeter which is intercepted between the foot of the sine and the arc,
uDA.
8. The tangent of an arc is a right line which touches it in one ex
tRmity, and is limited by a right line drawn from the centre of the
drde through the other extremity, as A T.
9. The tecant of an arc is the radial line which thus limits the
tugent, as C T.
10. These are also, by way of accommodation, said to be the sine,
tasgent, &c., of the angle measured by the aforesaid arc, to its deter
mintte radios.
11. The cosine of an arc or angle, is the sine of the complement
of that arc or angle, as G B : the cotangent of an arc or angle is the
tiDgent of the complement of that arc or angle, as E M. And the
OHxntd sine E O, and cosecant CM, are similarly the versed sine
and secant of the complement.
12. The suversine of an arc is the versed sine of its supplement,
MAD.
^flte.— The following contractions are employed to express the
foregoing terms, viz. : —
For the radius of the arc A B we write rad A B,
sine ditto sin AB,
tangent ditto tan AB,
secant ditto «^c A B,
versine ditto versin AB,
cosine ditto om AB,
cotangent ditto cot AB,
cosecant ditto cosec AB,
coversine ditto covers AB.
13' The /oiiowing Corollaries may he drawn from the above
Definitions: —
(A.) When the arc is evanescent, the sine, tangent, and versed
■Be^ are evanescent also, and the secant becomes equal to the radius,
that being its minimum limit. As the arc increases from this state,
the snes, tangents, secants, and versed sines increase ; thus they con
tnme till the arc becomes equal to a quadrant A £, and then the sine
K m its maximum state, being equal to the radius, and is then called
^ sine total; the versed sine is also then equal to the radius ; and
^ tecaot and taneent becoming incapable of mutually limiting each
<Hher, are regarded as infinite.
Id employing these lines for the purposes of calculation, they are
■D eonsidercKl as htmng positive values for any arc not exceeding 90®;
hit in the second quadrant, the cosine falling on the opposite side of
the diameter, and being measured in an opposite direction, is con
'i^ered negative^ but the sine remains positive; in the third quadrant
136
PLANE TRIGONOMETRY.
[part I.
the cosine is still negative^ and the sine, having now changed its direc
tion, is negative also ; in the fourth quadrant, the cosine having again
returned to that side of the diameter on which it was in the first
quadrant, again hecomes positive^ hut the sine remains neg€Uive. The
signs of the others are determined by the ordinary rules of algebra,
from the formulae at page 1 37. The following table exhibits both the
value of trigonometrical lines at the commencement of each qua
drant, and also the signs with which they are affected in passini
through the same.
Value
atO«.
Sign
in Ist
Quad.
Value
atOO*.
Sign
in 2nd
Quad.
+
1
Value'
atl80*.
_
O
Sign
in 3rd
Quad.
Value
at
270*.
Sign
in 4th
Quad.
Value
at
aeo*.
Sin ...
O
f
R
R
o
Tan...
O
f
00
—
o +
00
—
o
Sec ...
R
+
00

R —
00
+
R
Versin
O
+
R
■f
2R +
R
+
o
Cos ...
R
+
O

R
—
O
+
R
Cot ...
00
+
O
—
«
i
O
—
00
Cosec
00
+
R
f
00  1
R
—
00
R signifies equal to rad ; oo — infinite ; — evanescent.
(B.) An arc and its supplement have the same sine, tangent, anc
secant.
(C.) Of any arc less than a quadrant, the arc is less than its cor
responding tangent; and of any arc whatever, the chord is less thai
the arc, and the sine less than the chord ; but the smaller the arc, th<
nearer they all approach to equality.
(D.) The sine BD of an arc A B, is half the chord BF of th<
double arc B A F.
(£.) The versed sine of an arc, together ^4th its cosine, are equa
to the radius. Thus, AD fBG=AD + DC = AC.
(F.) The radius, tangent, and secant, constitute a rightangle<
triangle CAT; and the cosine, sine, and radius, constitute anothei
rightangled triangle C D B, similar to the former. So, again, the co
tangent, radius, and cosecant, constitute a third rightangled triangle
MEG, similar to both the preceding. Hence, when the sine anc
radius are known, the cosine is determined by the property of the
rightangled triangle. The same may be said of the determination ol
the secant, from the tangent and radius, &c. &c. &c.
(G.) Further, since the triangles CAT and MEC are similar
A T : CA : : B D : D C : : C E : E M, or tan ; rad : : sin : cos : : rad
: cot •
Also C T : C A : : C B : C D : : C M : £ M, or sec : rad : : rad
. : cosec : cot.
And CM:CE::CB:DB or cosec : rad : : rad : sin.
cos
CHAP, v.] PLANS TBIGONOMBTBY. 187
(H.) Also, by Geom. Sect. V. Prop. XXII., as AD : DB : : DB
: HD, or sin* = H D . versin ; or since H D = pad  cos, tben
sin^
tersin =
rad + cos
(I.) From Arts. (F.) (G.) and (H.) we deduce the following
fonnolsB :—
1. sin = >/Tad* — cos* = .
cosec
2. tan := >/sec* — rad* =
3. sec = v^rad^* + tan* =
cos
rad . sin rad^
cos cot
rad»
4. versin =
rad + cos *
5. cos = >/rad'^ — sin* =
sec
6. cot := V cosec' — rad""* =
rad«
c
rad . cos rad^
sin tan
7. cosec =: Vrad* + cot* = ;— .
sin
cos*
8. covers = •
rad + sin*
CK.) If unity be regarded as tbe radius of the circle, the above
fonnoijg become : —
1. sin = >/l — cos* = .
cosec
/ — 5 sin 1
2. tan = V sec* — 1 := — = —  .
cos cot
3. sec = >/l + tan*= — .
cos
sin*
4. versm =
1 + cos
5. cos = >/ 1 — sin* := — .
sec
cos
cot = V cosec? — 1 = r^ : —
sm tan
138 PL^NB TBIOONOMBTBY. [PABT I.
7» cosec = >/l + cot* = — r .
sm
COS*
8. covers = :.
1 + sm
14. The following are some of tbe most useful formulsB relating
to trigonometrical lines, in all of which radius is regarded as unity.
(a.) Eapressions/or the sine and cosine : — Let a equal tbe length
of the arc, then
"'"* = ''r:T:3 + 1.2.3.4.5  1.2.3.1.5 6.7 +^^'^
oosa=l^+P^^L^ ^^ /^^^ +&c (2).
(6.) Expressiofis for the sum and difference of two arcs: — Let
a and h be the two given arcs, then
sin (a 4 5) = sin a cos 6 + cos a sin b (3)
sin (a — 5) = sinacos b — cosasin b (4)
cos (a 4 &) = cos a cos 5 — sin asin b (5)
cos (a — ft) = cos a cos ft + sin a sin 6 (6)
, ,. tan a + tan ft .
tan(a f ft) = , ^ — t (7)
^ ' 1 ■— tan a tan ft ^
tana — tan ft
tan(aft = — — — j (8)
"^ I H tan a tan ft ^ '
cot a cot ft — i
cot(a + ft) = TTT^ — 7— (9)
^ ' cot ft + cot a ^ ^
„ cot a cot ft + 1 , ^
cot(a  ft) = r — (10).
"^ ' cot ft — cot a ^ '
(c.) Expressions for the sine and cosine of mtdtiple arcs: — Let
n be any integral number, then
sin «a = cosa{(2sina)» ^^(2sina)^ + (!!Z?K^i:D(2sina)^
 """!°.i.y"^ '"A^<^c) ,„)
cosnasz: J{(2sina)" — n(2sina)— * + ^ ? "l^V ^sina)^
n(n — 4)(« — 5V . , . ,
1.2 3 ^^(28ina)« + &c.} (12
CHAP, v.]
PLANl TBIOONOMBTBY.
139
(c^) Table o/mMkipk arcs : — Let « be the ain, t the tan, and e the
cot of the arc a, then
1. sin
2"V 2 >
2. 8ina=:«,
3. 8m2a = 2« v^l — «*,
4. m3a = Bs—4f\
5. 8in4a = (4* — 8r^ >/l — r*^
6. 8m5a = 16#' — 20*' + 5«,
13.tan = '_,
2 n.>/i^.^'
7. cos
2'
'5. tan 2a =
^^ Unda =
^^ tan4a =
^® tan 5a =
2/
13/**
4^ — 4/»
1— 6/» + /*'
1 — 10/« + 5^'
8. cos a ^ Vl — a",
9. cos 2a = 1 —2a',
10. cos 3a = (1 — 4«^) Vl— r',
11. co84a = l — 8*«^8«^
12. cos5a(l12**+16«*)Vr^.
« 1
19. cot = , ,
20. cot a =e,
21. cot2a=:
22. cot 3a =
23. cot 4a =
24. cot 5a =
c^1
c* — 3<?
3c« 1'
c*^6c'+ 1
4(r'4c '
c^— 10c3 i 5c
5c*— 10c« + 1 "
5s.
C ^') Table of Powers of the sine and cosine.
un as una,
Sain'aa 1 ..oo«2a,
liin^a"" Ssina — sin So,
Sun* a« 3 — 4 COS 2a 4 COS 4 a,
)6uii*as sin 6 a— 6 sin 3a4 lOsin a,
6. cos aBCOsa,
7. 2008*0* 0062a + I9
8. 4oos'a»co8 3a + 3oosa,
9. 8G08*a»oos4a + 4oos2a + 3,
10. 16co8*aB0066af5oos3ai10co8a.
». ^/) Expressions for the arc in terms of the sine or tangent : —
^^t « be the sin, and t the tan of the arc a, then
(13)
tKa = t —  + —6ie
(14).
SscT. II. Trigonometrical Tables.
From the forcing, and other properties, and theorems, mathe
ba¥e computed the lengths of the sines, tangents, secants,
140
PLANE TRIOONOMBTBY.
[p^
.t 1
and versed sines (assuming unity for the radius) corresponding to w
from 1 second of a degree, through all the gradations of magni^^ad
up to a quadrant, or 90^ ; and the results of the computation^^ t
arranged for use in tahles called Trigonometrical Tablet. As, B=ioi
ever, these quantities have to he carried to several places of decSL mi
in order to ohtain sufficiently accurate results, their use in cal ^cali
tions is attended with much lahour, and therefore it is usual to enr~^plc
their logarithms instead ; hut in this case the assumed radius is t^ce
as 10,000000000 instead of unity, since with the latter most o^^ th
quantities would he fractional^ and therefore have negative ch wuwt,
teristics, the use of which would be inconvenient, and is supers^e
by taking the radius as above.
Table IV. in the Appendix is such a table of the logarithiKBS <
the sines, cosines, tangents, cotangents, secants, and cosecants^ f<
every minute from 1 minute to 90 degrees, calculated to a radius <
10,000000000 as above. It will be observed that the headm^s *
the columns run along the tops of the pages as far as the 45th de^i"^
after which they return along the bottoms of the pages in contra
order, as below : —
sin
D.
cosec
tan
D.
cot
sec
D.
co«
cos
D.
sec
cot
D.
tan
cosec 1 D.
&C^
The reason of this will be apparent, if we only consider that the cc^
cot, or cosec of an arc, is the sin, tan, or sec of the complement ^
that arc. The intermediate columns, headed D, contain the diffe^
ences of the consecutive logarithms in the contiguous columns o '^
either side ; it will be seen that the same difference is common to thti^
sin and cosec, the tan and cot, and the sec and cos; since fron^
rad* rad*^
Art. (I.) page 137, sin = — —^ or rad^ = sin . cosec; tan = — ^
or rad ^ = tan . cot ; sec =
cosec
rad 2
cot
cos
or rad '^ = sec . cos ; and conse
quently, log sin + log cosec s= log tan 4" log cot = log sec + log
cos = 2 log rad = 20* ; therefore as the sin, tan, or sec increases,
so must the corresponding cosec, cot, or cos, diminish, and their dif
ferences must be equal.
USB OP TABLE IV.
To find the logarithmic sine^ tangent^ Spe, of a given are.
If the arc contains only degrees and minutes, its sin, tan, &c., will
be found simply by inspection, by looking along the top or bottom of
the tables for the degrees, and then in the first or last vertical column
for the minutes, according as the number of degrees is less or greater
than 45 ; and on the same line, in the column having for its title
.v.] PLANE TRIOONOMBTRT. 141
ler At the top or bottom, according as the degrees were found) the
z of the trigonometrical quantity required, its log will be found.
f the arc contains seconds as well, the logarithm must be found as
ife for the degrees and minutes ; then take the number in the
otigooas column headed D on the same line, multiply it by the
nnberof seconds, and divide by 100 (which is done by cutting off
le two list figures) ; the quotient must then be added to or subtracted
RND the log already taken out, according as the same would be in
aeued or decreased by an increase in the arc.
Examples,
1. Find the log sin of 37° 47'.
As the arc is less than 45°, by looking along the top of the table for
the degrees, and in the^r«/ column for the minutes, we find in the
colomn baying at its top the word sin, the figures 9*787232, which is
the log sin of the arc required.
2. Find the log tan of 75° 34'.
Here, as the arc is greater than 45°, looking at the battom of the
tihles for the degrees, and in the last column for the minutes, we find
in tbe column having tan at the bottom, 10*589431, the tan of
rS'' 34'.
3. Find the log sin of 31° 45' 5".
Tie log sin of 31° 45' is . . . . 9721 162 I
The No. in col. D is 3402 x 6" f 100 = + 1 70  10
.. The log sin of 31° 45' 6" = . 9721332
4. Find the log cos of 25° 1' 47".
Tbelogcos of 25° 1' is .... 995721 7
The No. in col. D is 983 x 47" h 100= ^ 462 01
•. The log cos of 25° 1' 47" = . . 9956755
Bb
Tojind the are corresponding to any given log sin, tan, &c. : —
^km Table IV. for the given log sin, &c., or the next less log
^''^'^ and on the same line will be found the minutes, and at the
^P or bottom of the page the degrees of the arc required ; if the log
^ foond is less than the given log, subtract the former from the
p^i add two cyphers to the right of the remainder, and divide it
7 the nnmber found in the contiguous column headed D ; the quo
^Qt will he the nnmber of seconds to be added to the degrees and
^tes in the arc already obtained.
Examples.
J* Fbd the arc wboae log tan is 10*577537.
Here the arc is found by inspection to be 75° 11'.
142 PLANB TBIQONOMBTBY. [PABT I.
2. Find the arc corresponding to the log sin 9*395401.
Given log == 9396401
Next less log = 9396166 = log tan 14*> 23'
23500 f 492 = 48 seconds ;
.. 9396401 is the log tan of 14° 23' 48",
In tbe sines and tangents of arcs less than about 2°, the differences
between any two successive values are so great (as will be seen by
an inspection of column D in tbe table), that the method above given
for finding the intermediate values for seconds will not be sufficiently
correct ; and the same remark applies to the cosines and tangents of
arcs greater than about 88°. It will also be observed, that in the
cosines and secants of arcs less than 2°, and in the sines and cosecants
of arcs greater than 88°, the differences are too small to enable us to
calculate accurately the value of any arc from them.
The first of these difficulties may be removed by the rules given
below for determining the values of the sines and tangents of small
arcs, and the tangents of large arcs, and conversely the arcs from the
sines and tangents. The second difficulty, however, could only be
got over by extending the tables to more decimal places, but as this
would also require all other Quantities employed in the same calcula
tions to be taken to an equal number of decimals, much additional
trouble would be occasioned ; and it is therefore better for determin
ing the value of an arc when near 90°, to employ some other function
than its sine, as, for instance, its cosine. In order to render this clearer
to those who are not familiar with the use of logarithms, we subjoin
an example of such a substitution of the cosine for the sine.
For instance, let it be desired from the formula,
Pj : Pg • : sin /9 : sin ^
to determine the value of the angle ^, when P j = 6001 ; P^ = 669;
and the angle = 63° 45'.
First by multiplving the means and extremes, and dividing both
sides by P, , we obtain
. . Pnsinfi
sm ^ = p (a).
Then from Art. (L), page 137,
sin ^ = >/ 1 — (cos iy
Pj sin $
= N/l.(cosd)«;
•^1
squaring both sides
/ p, sin e y
{^^ — J = 1  (cos ay,
CKJIP.?.] PLANB TBIOONOMBTRT. 143
trmssponng, and eztnctiiig the square root
cos
,=V(^)
(«)•
We haYe, therefore, two equations (a) and (^), from either of
'vrlieh we can ohtain the yalae of the angle ^, hy substituting the
iralQes of the known quantities ; but in doing so we shall find that
the second equation will give the yalue of J much more exactly than
tliefint.
Thua^by tubstitnting the known quantities in equation (a), we have
669 X sin 63° 45' . .
— — = sm i.
6001
Whence by logarithms.
Log sin ea** 45' =r 9952731
Log 669 = 2825426
12778157
Log 6001 = 2778224
Log sin i = 9999933
.. i = 88<> 59' 25".
Phiceedmg in a similar manner with equation (5), we have
/669 X sin 63'' 45'V
^C — 6001 — ;=<^^
Wbeoce by logarithms, taking the radius as unity, for the reasons
wpliined at page 146,
Log sin 63° 45' = 1952731
Log 669
= 2825426
Log 6001
2778167
= 2778224
T999933
2
Loff 99969
= r999866
Then I ^ 99969 = 00031, the log of which = 4491362,
and 4491362 r 2 = 2245681 = cos i;
or restoring the radius of the tables,
cos ^ = 8245681 = sin of the complement of i ;
144 PLINB TRIOONOMBTBY. [PABT I.
therefore, bj the rule given below for finding a small arc from its sin,
we have
8246681
6314425
22
3560128 = 363186 seconds = r 0' 31" 86 ;
^^=^ . ^ _ ggo 59/ 2g// .14^
whence we see that the former value of J obtained from equation
(a) is upwards of three seconds too small.
To find nccurately the log sin of an are less than 2°.
Reduce the arc to seconds, and find the log of that number from
Table III., to which add 4685575 (the log sin of 1'), and subtract
onethird of the decimal portion of the log sec of the arc taken from
Table IV. ; the remainder is the log sin of the arc required.
Examples.
Find the log sines of 13' and of 1° 3'.
Log (13' X eW) = 780"  2892096
4685575
7577670
Log8ecofl3'»000003r3» 000001
Log sin of 13' « 7577669
Log (ey X 60) « 3780" « 3577492
4685575
8263067
Logsecof r3'»000073^3» 000024
Log sin of 1° y ^ 8263043
To find accurately the log tan of an arc less than 2®.
To the log of the number of seconds in the arc add 4685575, and
twothirds of the decimal portion of its log sec; the sum is the log tan
of the arc required.
Examples,
Find the log tans of 24' and of 1° l.V.
Log (24' X 60} : 1440" <= 3158363
4085575
Log8ecof24'«*000011x}" '000007
Log tan of 24' « 7843945
Log (75^ X 60)  4500" » 3653213
4685575
Log sec of ri5'»=000l03x » 000068
Log Un of r 15^ « 8338856
To find accurately the log tan of an arc greater than about 88®.
Add to the log of the number of seconds that the arc is less than
90% twothirds of the decimal portion of the log cosec, and subtract
the sum from 15*314425; the remainder will be the log tan required.
Examples.
1. Find the log tan of 89° 5' 13".
90'' 0' 0"
89 5 13
0^ 64' 47" = 3287 seconds
CBAP.V.] PLANS TRIOONOMBTBY. 145
Constant log .... = 15314426
Log of 3287 . . . ; = 36 16800
Log cosec of 89« 5' 13'' = 000057 x  = 000038
3516838
Log tan of 89'' 5' 1 3'' = 11797587
a. Find the log tan of SS'' 61' 10".
90° 0' 0"
88 51 10
r 8' 50" = 4130 seconds.
CoMtMtlog =15314425
Ii0gof4130 =3615950
Xogcoscc of 88** 51' 10" = 000087 x  = 000058
3616008
Log tan of 88° 61' 10" = 1169 8417
To find accurately an arc of not more than il°/rom its log sine,
Totbegiyen log sin, add 5*314425, and onethird of the decimal
poition of the secant of the nearest arc to that whose log sin is given,
^« som, rejecting 10 from the characteristic, will be the logarithm
^^ tke number of seconds in the arc.
Example.
Heqnired the arc whose log sin is 8314719,
GiTen log sin . . . = 8314719
Constant log .... = 5314425
Ug sec of nearest arc = 000093 f 3 = 000031
Arc required 1° 10' 58" = 4258" = 3629175
To find accurately an arc of not more than ^^from Us log tan,
To the giTcn log tan add 5*314425, and from the sum subtract
^''^[tJWs of the decimal portion of the log sec of the arc whose log
^i« nearest to that given, and the remainder, rejecting 10 from the
^''ctemtic, will be the log of the number of seconds in the arc.
Example,
Required the arc whose log tan is 8*231461.
Given log tan . . . . = 8*231461
Constant log . . . . = 6*314425
13545886
Log sec of nearest arc = 000063 x  = 0000 42
Required arc = 58' 34" = 3514" = 13546844
L
146 GENEBIL PROPOSITIONS. [PABT I.
To find (accurately an arc greater than S^"^ from its log tan.
Add to the given log tan twothirds of the decimal portion of the
log cosec of the nearest arc to that whose log tan is given, and sub
tract the sum from 15*314425, the remainder is the log of the num
ber of seconds that the arc is less than 90^.
Example,
Required the arc whose log tan is 11*695900.
Constant log ... . = 15814425
Given log tan .... =11695900
Log cosec of nearest arc 000088 X  = 000059
11695959
Required arc = 1° 9' 14'' = 4154" = 3618466
ON THE ROOTS AND POWERS OF TRIGONOMETRICAL QUANTITIES.
In extracting the root, or raising the power of any trigonometrical
quantity by means of its logarithm, it will always be found most con
venient to reduce the assumed radius to unity, by subtracting 10 from
the characteristic of the logarithm, observing where the characteristic
thus becomes negative the rules given at pages 47 and 48 An ex
ample of this alteration of the charactenstic is given at page 143.
TRIGONOMETRICAL QUANTITIES OP ANGLES GREATER THAN 90*^.
Although Table IV. only purports to give the sines, tangents, &c.,
of angles less than, or equal to OO**, any of these functions of angles
greater than 90° may readily be obtained, since any function of an
angle greater than 90° is equal to the same function of the supple
ment of that angle ; due regard being had to changing the sign, if
requisite, according to the table given at page 136. Thus the log sin
of 141° 15' is 979052 1, the same as the sin of its supplement
38° 45'; the log sec of 95° 43' is — J1001701, or the sec of 84° 17'
with its sign changed ; the log cosec of the same is I000S165, or the
cosec of 84° 1 7', the sign remaining the same ; and the log tan of
173° 4' is — 9 084947, or the tan of 6° 56', with its sign altered.
Sect. III. General Propositions,
I. The chord of any arc is a mean proportional between the versed
sine of that arc and the diameter of the circle.
II. As radius is to the cosine of any arc, so is twice the sine of
that arc to the sine of double the arc.
III. The secant of any arc is equal to the sum of its tangent, and
the tangent of half its complement.
IV. The sum of the tangent and secant of any arc, is equal to the
tangent of an arc exceeding that by half its complement. Or, the
/
CfliP. v.] OBNBBAL PROPOSITIONS. 147
9am of the tangent and secant of an arc is eqoal to the tangent of
45'/>/i«half the arc,
V. The chord of 60^ is eqaal to the radius of the circle; the
^vereed sine and cosine of 60° are each equal to half the radius, and
the secant of 60° is equal to double the radius.
VI. The tangent of 45° is equal to the radius.
VII. The square of the sine of half any arc or angle is equal to
a rectangle under half the radius and the versed sine of the whole
arc; and the square of its cosine is equal to a rectangle under half
tiie radios and the versed sine of the supplement of the whole arc or
aogle.
nil. The rectangle under the radius and the sine of the sum or
tile difference of two arcs is equal to the sum or the difference of the
i^Bctangles under their alternate sines and cosines.
IX. The rectangle under the radius and the cosine of the sum or
^e difference of two arcs, is equal to the difference or the sum of
the rectangles under their respective cosines and sines.
X. As the difference or sum of the square of the radius and the
'^^ctaogie under the tangents of two arcs, is to the square of the
'^xiios; so is the sum or difference of their tangents, to the tangent of
t^ sum or difference of the arcs.
•XI. As the sum of the sines of two unequal arcs, is to their dif«
^f^nce; so is the tangent of half the sum of those two arcs to the
^Sigent of half their difference.
^11. Of any three equidiffereut arcs, it will be as radius is to the
^**«ne of their common difference, so is the sine of the mean arc, to
"^If the sum of the sines of the extremes ; and, as radius is to the
1^X1 c of the common difference, so is the cosine of the mean arc to
"^If the difference of the sines of the two extremes.
4^A.) If the sine of the mean of three equidiffercnt arcs (radius
■^^ing unity) be multiplied into twice the cosine of the common dif
^^*^ce, and the sine of either extreme be deducted from the pro
^'ict, the remainder will be the sine of the other extreme.
(B.) The sine of any arc above 60°, is equal to the sine of an
other arc as much below 60°, together with the sine of its excess
•^Tc 60°.
^. From this latter proposition, the sines below 60° being known,
**^<»e of arcs above 60** are determinable by addition only.
^ni. In any rightangled triangle, the hypothenuse is to one of
he legs, ag the radius is to the sine of the angle opposite to that
^^; and one of the legs is to the other as the radius is to the tan
S^t of the angle opposite to the latter.
. ^IV. In any plane triangle, as one of the sides is to another, so
^ the sbe of the angle opposite to the former to the sine of the
^^^ opposite to the latter.
^^* In any plane triangle it will be, as the sum of the sides
Jjj^t the vertical angle is to their difference, so is the tangent of
'^ the sam of the angles at the base, to the tangent of half their
«wrence.
l2
148 SOLUTION OP THK CASES OP PLANE TBIANOLBS. [PABT I.
XVI. In any plane triangle it will be, as the cosine of the dif
ference of the angles at the base, is to the cosine of half their sum,
so is the sum of the sides about the vertical angles to the third side.
Also, as the sme of half the difference of the angles at the base, is
to the sine of half their sum, so is the difference of the sides about
the vertical angle to the third side, or base *.
XVII. In any plane triangle it will be, as the base is to the sum
of the two other sides, so is the difference of those sides to the dif
ference of the segments of the base made by a perpendicular let fall
from the vertical angle.
XVIII. In any plane triangle it will be, as twice the rectangle
under any two sides, is to the difference of the sum of the squares of
those two sides and the square of the base, so is the radius to the
cosine of the angle contained by the two sides.
Cor, When unity is assumed as radius, then if A C, A B, B C, are
the sides of a triangle and C the angle opposite the side A B, this
AC^ h BC^ — AB , . .,
Prop, gives cos C = —  : and similar expressions
<« C B . C A
for the other angles.
XIX. As the sum of the tangents of any two unequal angles is to
their difference, so is the sine of the sum of those angles to the sine
of their difference.
XX. As the sine of the difference of any two unequal angles is
to the difference of their sines, so is the sum of those sines to the
sine of the sum of the angles.
These and other propositions are the foundation of various for
mnlsB, for which the reader who wishes to pursue the inquiry may
consult the best treatises on Trigonometry.
Sect. IV. Solution of the C<we« of Plane Triangles,
Although the three sides and three angles of a plane triangle,
when combined three and three, constitute twenty varieties, yet they
furnish only three distinct cases in which separate rules are required.
CASE I.
When a side and an angle are two of the given parts.
The solution may be effected by Prop. XIV. of the preceding sec
tion, wherein it is affirmed that the sides of plane triangles are re
spectively proportional to the sines of their opposite angles.
In practice, if a side be required, begin the proportion with a sine,
and say,
As the sine of the given angle.
Is to its opposite side ;
So is the sine of either of the other angles,
To its opposite side.
* Thene propositions were first given by Thacker in his Mathematical Miaeel
lanjfy published in 1743 ; their practical utility haM been recently shown by Pro
feMvr Wallac€t in the Edinburgh Philosophical TraruacHons.
AV.Y.] SOLUTION OF TUB CASKS OF PLANE TRIANGLES. 149
If an M^ he required, begin the proportion with a side, and
As one of the given sides,
Is to the sine of its opposite angle ;
So is the other given side,
To the sine of its opposite angle.
The thud angle becomes known by taking the sum of the two
former from 180°.
NUe. — It is usually best to work the proportions in trigonometry
bj means of the logarithms, taking the logarithm of the Jirst term
from the sum of the logarithms of the 9econd and thirds to obtain the
logarithm of the fourth term. Or, adding the arithmetical comple
ifiAi/of the logarithm of the first term to the logarithms of the other
two, to obtain that of the fourth.
CASE II.
When two sides and the included angle are given.
The solution may be effected by means of Props. XV. and XVI.
of die preceding section.
Thus: take Uie given angle from 180% the remainder will be the
sum of the other two angles.
TlienMy,
As the sum of the given sides,
Is to tlieir difference ;
So is the tangent of half the sum of the remaining angles.
To the tangent of half their difference.
Then, secondly say,
As the cosine of half the said difference.
Is to the cosine of half the sum of the angles ;
So is the sum of the ffiven sides.
To the third, or required side.
As the sine of half the diff*. of the angles,
Is to the sine of half their sum ;
So is the difference of the given sides,
To the third side.
^•tmjU.^hi the triansle ABC (Fig. 9.) are given A C = 450,
^C 3 540, and the included angle C c= 80"* ; to find the third side,
^ the two remaining angles. Then,
Log(BC AC as 90)= 1954243
Log tan (i A + B = 50<>) = 10076187
12030430
Log(BCf'AC =: 990) = 2995685
Log tan (4 A  B =c 6^ 11") = 9034795
150 SOLUTION OF THE CASES OF PLINE TRIANGLES. [PlBT I.
Then, Log cos ( A i B = 50°) = 9808068
Log (BC h A C = 990) = 2995635
12803703
Log cos (i A — B = 6° 1 V) = 9997466
Log(AB ... =64008)= 2806237
Also, ^(A iB) + HAB) = 56° ir=A; and J (A f B)
_ J (A  B) = 43° 49' = B.
Here, much time will be saved in the work by talcing log cos J
(A h B) from the tables, at the same time with log tan ^ (A h B) ;
and log cos (A — B) as soon as log tan ^ (A — B) is found. Ob
serve, also, that the log of B C I A C is the same in the second
operation as in the first. Thus the tables need only be opened in
Jive places for both operations.
A nother solution to Case II,
Supposing C to be the given angle, and C A, C B, the given sides ;
then the third side may be found by this theorem, viz. ; —
A B = ^/(A C^ f B C'' — 2 A C . C B . cos C).
Thus, taking A C = 450, B C = 540, C = 80% its cos 1736482
AB = ^(450^^+ 540'^ 2 x 450 x 540 x 1736482)
= ^{90=^ (5 + 6' 2 X 5 X 6 X 1736482)}
= 90 ^5058118 = 90 X 7112 = 64008, as before.
CASE III.
When the three sides of a plane triangle are given^ to find the angles,
1st Method. — Assume the longest of the three sides as base, then
say, conformably with Prop. XVI., last section,
As the base.
Is to the S!im of the two other sides ;
So is the difference of those sides.
To the difference of the segments of the base.
Half the base added to the said difference gives the greater seg*
ment, and made less by it gives the less ; and thus, by means of the
perpendicular from the vertical angle, divides the original triangle
into two, each of which falls under the first case.
2nd Method. — Find any one of the angles by means of Prop. XVIII.
of the preceding section ; and the remaining angles either by a repe
tition of the same rule, or by the relation of the sides to the sines of
their opposite angles, viz. : —
A C^ + B C2 — A B* A B'^ + B C — A C*
cos C = ; cos B = ir7~——^
2AC.BC 2AB.BC
B A^ + A C^ — B C«
and cos A = .
2AB . AC
r^HAP. v.] SOLUTION OP THB CASES OF PLANA TRIANGLES. 151
RIGHTANGLED PLANE TRIANGLES.
Rightangled triangles may (as well as others), be solved by means
if the rule to the respecUye case under which any specified example
alls; and it will then be found, since a right angle is always one of
lie data, that the rule usually becomes simplified in its application.
When two of the sides are given, the third may be found by
neans of the property in Geom.^ Prop. XVI. Sect. III.
Hypoth. = v^fbase^  P^rp^)
Base = >/(hyp.* — perp.*) = >/(hyp. + perp.) . (hyp. — perp.)
Pcrp. = ^(hyp.^ — base^) = >/(hyp. + base) . . (hyp. — hase).
There is another method for rightangled triangles, known by the
phnse making any Me radius ; which is this.
** To find a side. Call any one of the sides radius, and write upon
it the word radius ; observe whether the other sides become sines,
tansents, or secants, and WTite those words upon them accordingly.
Call the word written upon each side the name of each side:
then say,
As the name of the given side,
Is to the given side ;
So is the name of the required side,
To the required side."
^ To find an angle. Call either of the given sides radius, and
vrite upon it the word radius; observe whether the other sides
^^^come sines, tangents, or secants, and write those words on them
accordingly. Call the word written upon each side the name of that
"We. Then say,
As the side made radius.
Is to radius ;
So is the other given side.
To the name of that side,
vhich determines the opposite angle."
When the numbers which measure the sides of the triangle are
<i^ nnder 12, or resolvable into factors which are each less than
1^) the solution may be obtained, conformably with this rule, easier
vithoQt logarithms than with them. For,
Ut ABC (Fig. 103) be a rightangled triangle, in which A B,
the hose, is assumed to be radius ; B C is the tangent of A, and A C
te secant, to that radius ; or, dividing each of these by the base, we
"ifl ha?e the tangent and secant of A, respectively, radius being
^tj. Tracing in like manner the consequences of assuming BC
(tt fig. 104), and AC (as fig. 105), each for radius, we shall readily
•l>tMn these expressions.
1. ^p^* = tan angle at base. (Fig. 103.)
2. ^ = sec angle at base. (Fig. 103.)
base
152 HEIGHTS AND DISTANCES. [PART
3. = tan angle at vertex. (Fig. 105.)
4. ^^ = sec angle at vertex. (Fig. 105.)
5. r—^' = sin angle at base. (Fig. 104.)
base _,
6. = sm angle at vertex. (Fig. 104.)
Sect. V. On the application of Trigonometry to measuring HeighU
and Distances.
Trigonometry receives its principal practical application in the
operations of surveying, and measuring heights and distances; as,
however, the methods of its application (depending on the peculiar
circumstances of each case) are exceedingly various, we cannot lay
down any general rules, but must content ourselves with giving a
selection of such examples as are most likely to occur ; and the prin
ciples developed in which, will be sufficient to guide any person in its
further application to other cases.
The instruments employed to measure angles are quadrants, sex
tants, theodolites^ &c., the use of either of which may be sooner
learnt from an examination of the instruments themselves than from
any description independently of them. For military men and for
civil engineers, a good pocket sextant, and an accurate micrometer
(such as Cavallo's) attached to a telescope, are highly useful. For
measuring small distances, as bases, 50 feet and 100 feet chains, and
a portable box of graduated tape, will be necessary.
For the purposes of surveying, it is usual to employ a chain 66 feet
in length, subdivided into 100 links, each 7*92 inches; the reason for
using a chain of this length is, that ten of such square chains are
equal to an acre, and therefore the acreage of the several divisions
of an estate are found with much greater facility when measured in
chains and links, than when the measurements are taken in feet.
Eofample 1.
In order to find the distance between two trees, A and B (Pig. 9),
which could not be directly measured because of a pool which occu
pied much of the intermediate space, I measured the distance of each
of them from a third object, C, viz., A C = 588, B C = 672, and
then at the point C took the angle ACB between the two trees
:= 55° 40'. Required their distance.
This is an example to Case II. of plane triangles, in which two
sides, and the included angle, are given. The work, therefore, may
exercise the student: the answer is 593*8.
CHiP.V.] HEIOHTS AND DISTANCES. 153
Example 2.
Wanting to know the distance between two inaccessible objects,
C and D (Fig. 106), wbicb lay in a direct line from the bottom of a
tower on whose top I stood, I took the angles of depreuion of the
two objects, viz., of the most remote 25 J°, of the nearest 57°.
^4t is the distance between them, the height of the tower A B
being 120 feet?
HAD = 25' 30', hence BAD = BAH — HAD = 64''30'.
flAC=57'' O', hence BAG = BAH — H AC =33'' 0'.
Hence the following calculation, by means of the natural tangents.
^OT^ if A B be regarded as radius, B D and B C will be the tangents
of tke respective angles BAD, BAC, and CD the difference of
^ose tangents. It is, therefore, equal to the product of the difference
»f tie natoral tangents of those angles into the height A B.
Thus, nat. tan 64° 30' = 20965436
nat tan 33° = 0*6494076
difference 14471360
multiplied by height, 120
gives distance CD = 173*6563200
*/ The natural sines, tangents, &c., are easily obtained from
^^^We IV., by subtracting 10 from the characteristics, and then
^iifiing the natural number answering to the logarithms with their
^^^^•neteristics so altered.
Example 3.
Standing at a measurable distance AB (Fig. 103), on a hori
^t>l pltne, from the bottom of a tower, I took the angle of eleva
^ of the top (C) ; it is required from thence to determine the
^l of the tower.
In this case there would be given A B and the angle A, to find
*C=:ABx tan A.
B? logarithms, when the numbers are large, it will be log B C =
^AB + logtanA.
^«..If angle
J A=ll° 19'
then B C = ^ A B very nei
»
A = 16 42
BC=AAB
5>
A = 21 48
BC= 1 AB „
»
A = 26 34
B C = ^ A B „
99
A = 30 58
B C = 1 A B „
>9
A = 35
BC = 375AB
99
A = 38 40
BC= 1 AB „
99
A = 45
B C = A B, ewacdy.
To HYe the time of computation, therefore, the observer may set
^ inttnunent to one of these angles, and advance or recede, till it
154 HEIGHTS AND DI8TAKCSS. [PART I.
accords with the angle of elevation of the object; its height above the
horizontal level of the observer's eye will at once be known, by
taking the appropriate fraction of the distance A B.
Example 4.
Wanting to know the height of a church steeple, to the bottom of
which I could not measure on account of a high wall between me
and the church, I fixed upon two stations at the distance of 93 feet
from each other, on a horizontal line from the bottom of the steeple,
and at each of them took the angle of elevation of the top of the
steeple, that is, at the nearest station 5b° 54', at the other 33° 20^.
Required the height of the steeple.
Recurring to figure 106, we have given the distance C D, and the
angles of elevation at C and D. The quickest operation is by means of
the natural tansrents, and the theorem A B = .
® ' cot D — cot C
Thus cot D = cot 33° 20' = 1*5204261
C = cot 55 54 = 6770509
Their difference = 8433752
93
"^"•"''^" = i433752 = "«'''"^'
Eaoample 5.
Wishing to know the height of an obelisk standing at the top of
a regularly sloping hill, I first measured from its bottom a distance of
36 feet, and there found the angle formed by the inclined plane and
a line from the centre of the instrument to the top of the obelisk
41°; but after measuring on downward in the same sloping direction
54 feet farther, I found the angle formed in like manner to be only
23° 45^ What was ttie height of the obelisk, and what the angle
made by the sloping ground with the horizon ?
The figure being constructed (see fig. 107), there are given in
the triangle ACB, all the angles and the side AB, to find BC.
It will be obtained by this proportion, as sin C (= 17° 15'= B — A)
: AB (= 54) : : sin A(= 23° 45') : BC = 733392. Then, in
the triangle D B C are known B C as above, B D = 36, C B D =
41°; to find the other angles, and the side CD. Thus, first, as
CB f BD : CB — BD : : tan(D f C) = ^(139°) : tanJ(D — C)
= 42° 24 J'. Hence 69° 30' + 42° 24' =112° 54 J' = C D B, and
69°30' — 42°24' = 27°5J' = BCD. Then, sin BCD : BD ::
sinCBD : CD = 51 '86, height of the obelisk.
The angle of inclination DAE = HDA = CDB — 90° = 22° 54i'.
Remark. — If the line BD cannot be measured, then the angle
D A E of the sloping ground must be taken, as well as the angles
C A B and C B D. In that case D A E + 90° will be equal to C D B:
HEIGHTS AND DISTANCES. 155
ifter C B is found from the triangle A C B, CD may be
the triangle C B D, by means of the relation between the
. the sines of their opposite angles.
Example 6.
; on a horizontal plane, and wanting to ascertain the height of
standing on the top of an inaccessible hill, I took the angle
•tion of the top of the hill 40% and of the top of the tower
en measuring in a direct )ine 1 80 feet farther from the hill, I
1 the same vertical plane the angle of elevation of the top of
wer 33^ 45^. Required from hence the height of the tower.
e figure being constructed (see fig. 108), there are given, AB
«0, C A B = 33° 45', ACB = CBE — CAE=17°15', CBD
r, BDC = 180° — (90° — DBE)=130°. And CD may be
ad from the expression C D . rad^ = A B . sin A . sin C B D . cosec
:B.8ecDB£.
Or,
Dsing logarithms.
log
AB
=
180»
=
2255273
+ log sin
A
=
33°
45'
=
9744739
f log sin
CBD
=
11°
0'
=
9280599
f log cosec
:ACB
=
17°
15'
=
10527914
f log sec
DBE
*"~
40°
0'
"~~
10115746
41924271
]
log CD
=
log rad^
839983
=
40000000
1 924271
Example 7.
In order to determine the distance between two inaccessible ob
KcttAand B (Fig. 108), on a horizontal plane, we measured a con
TenieDt base, C D, of 536 yards, and at the extremities C and D took
^following angles, via., D C B = 40^ 16', B C A = 57" 40', C D A
=5 KV 22', A D B = 71° 7'. Required the distance A B.
Rnt, in the triangle C D A are given all the angles, and the
>^CD to find AD. So, again, in the triangle CDB, are given
iQtlietDgles, and CD to find DB. Lastly, in the triangle DAB
^ giren the two sides A D, B D, and the included angle A D B, to
fad A B=i 93952 yards.
£mari.~ In like manner the distances taken two and two, be
^*^ soy number of remote objects posited around a convenient
station line, may be ascertained.
Example 8.
Soppote that in carrying on an extensive survey, tbe distance be
^ecn two spires A and B (Pig. 109) has been found equal to 6594
1^*^ and that C and D are two eminences conveniently situated
for eitending ibe triangles, but not admitting of the determination
156 HEIGHTS i\ND DISTANCB8. [PABT I.
of tlicir distance by actual admeasurement: to ascertain it, therefore,
we took at C and D the following angles, viz. : —
CA C B = 85° 46' JA D C = 31° 48'
(B C D = 23° 5G' a D B = 68° 2'
Required C D from these data.
In order to solve this problem, construct a similar quadrilateral
Acdhy assuming cd equal to 1, 10, or any other convenient number:
compute A h from the given angles, according to the method of the
preceding example. Then, since the quadrilaterals Acdh^ A C D B,
are similar, it will be, as Ah : cd i: AB : Cl>; from which C D is
found to be equal to 4694 yards.
Example 9.
Given the angles of elevation of any distant object, taken at three
places in a horizontal right line, which does not pass through the
point directly below the object; and the respective distances between
the stations; to find the height of the object, and its distance from
either station.
Let A EC (Fig. 110) be the horizontal plane, FE the perpen
dicular height of the object above that plane, A, B, C, the three
places of observation, FAE, FBE, FCE, the angles of elevation,
and A B, B C, the given distances. Then, since the triangles A E F,
BEF, CEF, are all right angled at E, the distances A£, BE, CE,
will manifestly be as the cotangents of the angles of elevation at A,
B, and C.
Put AB = D, BC = </, EF = a?, and then express algebraically
the theorem given in Geom. Sect. III. Prop. XXV., which in this
case becomes,
AE2.BChCE.AB = BE2.ACfAC.AB.BC.
The resulting equation is
dx^^QOiAf 4 Da?^(cotC) = (D + rf)ar^(cotB)« + (D + d)l}d.
From which is readily found
(Df fl?)P<^
rf(cot A)2 + D(cotC)^  (D + <3?) (cot Bf '
Thus £ F becoming known, the distances A E, BE, C £, are
found, by multiplying the cotangents of A, B, and C, respectively,
by EF.
Remark. — When D=</, or Dfrf=2D = 2</, that is, when
the point B is midway between A and C, the algebraic expres
sion becomes,
a? = </i. >/ 1 (cot A)* h I (cot cy — (cot By,
which is tolerably well suited for logarithmic computation. The rule
inay, in that case, be thus expressed.
Double the log cotangents of the angles of elevation of the ex
treme stations, find the natural numbers answen'ng thereto, and take
half their sum ; from which subtract the natural number answering
CHAP, v.] HSIOHTS AND DISTANCES. 157
to twice the log cotangent of the middle angle of elevation : then
haiftbelogof this remainder subtracted from the log of the mea
sured distance between the first and second, or the second and third
stations, will be the log of the height of the object.
The distance from either station \%ill be found as aboye.
NtU.The case explained in this example, is one that is highly
useful, and of frequent occurrence. An analogous one is when the
angles of elcTation of a remote object are taken from the three angles
of a triangle on a horizontal plane, the sides of that triangle being
known, or measurable : but the above admits of a simpler computa
tion, and may usually be employed.
Example 10.
From a convenient station P (Fig. Ill), where could be seen tliree
objects, A, B, and C, whose distances from each other were known
(ra. A B = 800, A C =600, B C = 400 yards), I took the horizon
til ingles A P C = 33° 45', B P C = 22° 30'. It is hence required
to detemine the respective distances of my station from each object.
Here it will be necessary, as preparatory to the computation, to
descnbe the manner of
0(Mttnteti(m, — Draw the given triangle ABC from any convenient
Kile. From the point A draw a line A D to make with A B an angle
eqoal to 22° 30', and from B a line BD to make an angle DBA =
33^45'. Let a circle be described to pass through their intersection
D, ud through the points A and B. Through C and D draw a right
line to meet the circle again in P: so shall Pbe the point required.
For, drawing PA, PB, the angle A PD is evidently = AB D, since it
^ds on the same arc A D : and for a like reason B P D = B A D.
So that p is the point where the angles have the assigned value.
The result of a careful construction of this kind, upon a goodsized
««lc, will give the values of P A, P C, P B, true to within the 200th
pwtofeach.
Manner of Computation, — In the triangle ABC, where the sides
*n known, find the angles. In the triangle A B D, where all the
iiglesare known, and the sides A B, find one of the other sides A D.
We B A D from B A C, the remainder, D A C, is the angle included
Ween the two known sides, AD, AC; from which the angles ADC
ttdACD may be found, by Case II., p. 149. The angle C A P =
W  (A P C + A C D). * Also, BCP = BCA — ACD; and
'BC = ABC H PBA=:ABCf sup. ADC. Hence, the
^^ required distances are found by these proportions. As sin
APC: AC :: sin PAC : PC :: sin PC A : PA; and lastly, as sin
*PC : BC : : sin B P C : B P. The results of the computation are,
'A= 70933, PC = 104266, PB = 934 yards.
V The computation of problems of this kind, however, may be
1 litde shortened by means of an analytical investigation. Those
^ wish to pursue this department of trigonometry may consult the
•"•*»es by Bmnycattky Gregory ^ and WwdhoxMe,
158 HSIOHTS iIND DISTANCES. [PART I.
Noie. — If C had been nearer to P tlian A B, the general principles
of constmction and compotation would be the same; and the modi
fication in the process Terr obyioas.
DBTSEMINATION OF HSIOHTS AND DISTANCES BY APPBOXIMATS ME
CHANICAL METHODS.
1. For HeigkU.
1 . By sAadatPS^ when the son shines. — Set op yertically a staff of
known length, and measure the length of its shadow upon a horizon
tal or other plane ; measure also the length of the shadow of the ob
ject whose height is required. Then it w\]\ be, as the length of the
shadow of the staff, is to the length of the staff itself; so is the
length of the shadow of the object, to the object's height.
2. By two rods or staves set up vertically: — Let two staves, one,
say, of 6 feet, the other of 4 feet long, be placed upon horizontal
circular or square feet, on which each may stand steadily. Let A B
(Fig. 112) be the object, as a tower or steeple, whose altitude is
required, and A C the horizontal plane passing through its base. Let
C D and E F, the two rods, be placed with their bases in one and the
same line C A, passing through A the foot of the object; and let tliem
be moved nearer to, or farther from, each other, until the summit
B of the object is seen, in the same line as D and F, the tops of the
rods. Then bv the principle of similar triangles, it will be, as D H
(= C E) : F H : : D G (= C A) : BG ; to which add A G = C D, for
the whole height A B.
3. By reflection, — Place a vessel of water upon the ground, and
recede from it, until you see the top of the object reflected from the
smooth surface of the liquid. Then, since by a principle in optics,
the angles of incidence and reflection are equal, it will be as your
distance measured horizontally from the point at which the reflection
is made, is to the height of your eye above the reflecting surface ; so
is the horizontal distance of the foot of the object from the vessel to
its altitude above the said surface *.
4. By means of a portable barometer and thermometer, — Observe
the altitude B, of the mercurial column, in inches, tenths, and hun
dredths, at the bottom of the hill, or other object whose altitude is
required ; observe, also, the altitude, 6, of the mercurial column at the
top of the object ; observe the temperatures on Fahrenheit's thermo
meter, at the times of the two barometrical observations, and take the
mean between them.
* Leonard Digge§, in his curious work, the PatUometria^ published in 1571 «
first proposed a method for the determination of altitudes by means of a geo
metrical square and plummet, which has been described by various later authors,
as Ozanam, Donn, Hutton, &,c. But, as it does not seem preferable to the
methods above given, I have not repeated it here.
r.] HEIGHTS AND DISTANCES. 159
B — i
55000 X = height of the hill, in feet, for the tempera
of SS"" on Fahrenheit. Add ^ Jg of this result for every degree
d) the mean temperature exceeds 55°; suhtract as much for every
ree below 55°.
rhis will he a good approximation when the height of the hill is
« dian 2000 feet ; and it is easily remembered, because 55°^ the
flomed temperature, agree with 55, the significant figures in the co
fident; while the significant figures in the denominator of the cor
.•sctiDg fraction are two /burs,
5. Bifan extension of the principle of page 153.— Set the sextant,
»r other instrument, to the angle 45°, and find the point C (Fig. 103)
>zi the horizontal plane, where the object A B has that elevation : then
set ibc bstrument to 26® 34', and recede from C, in direction BCD,
kill the object has that elevation.
7% Munce C D between the two stations will ^ = A B.
So,sgain, if C = 40%D = 24°3lJ', CD will be = A B.
or, „ if C = 35% D = 22° 23', CD „ = A B.
or, „ if C = 30°, D = 20° 6', C D „ = A B.
or, „ if C = 20°, D = 14° 5e\ CD „ = A B.
or, generally, if cot D — cot C = rad, CD „ = A B.
^ For deciation from lend. — Let E represent the elevation of the
^^i>^t line to the earth above the true level, in feet and parts of a
'^K D the distance in miles: then E =  D'^
Tbis gives 8 inches for a distance of one mile ; and is a near ap
P*^ximauoD when the distance does not exceed 2 or 3 miles.
2. For Distances,
1* By means of a rhombus set off upon a horizontal plane, — Sup
P^O(Fig. 113) the object and O B the required distance. With
^ ibe or measuring tape, whose length is equal to the side of the in
^M rhombus, say 50 or 1 00 feet, lay down one side B A in the
direction BO towards the object, and BC another side in any con
'^taJeni direction (for whether B be a right angle, or not, is of no
^^'''^(tliience) ; and put up rods or arrows at A and C. Then fasten
^ ends of two such lines at A and C, and extend them until the
two other ends just meet together at D ; let them lie thus stretched
*pOD the ground, and they will form the two other sides of the
"•^^inbas AD, CD. Fix a mark or arrow at R, directly between C
^ 0, upon the line A D ; and measure R D, R A upon the tape.
'^ it will be as R D : D C : : C B : B O, the required distance.
OdksTwise, To find the length of the inaccessible line Q R.
At some convenient point B (Fig. 114) lay down the rhombus
J^ADC, 10 that two of its sides, B A, B C, are' directed to the ex
^''"Wwi of the line Q R. Mark the intersections, O and P, of A R,
160 HEIGHTS AND DISTANCES. [PART I.
C Q, with the sides of the rhomhus (as in the former method) : then
the triangle O D P will he similar to the triangle R B Q ; and the in>
accessihle distance R Q will be found = — .*
OD X DP
Thus,ifBA =BC,&c. =100fk.,OD = 9 ft. 5 ins.,D P = 11 ft.
10ins.,OP = 13 ft. 7 ins., then QR = ^^t^^^~^ = 1219 feet.
2. By means of a micrometer attached to a telescope. — Portable in
struments for the purpose of measuring extremely small angles, hare
been invented by Martin, Cavallo, Dollond, Brewster, and others. In
employing them for the determination of distances, all that is neces
sary in practice is to measure the angle subtended by an object of
known dimensions, placed either vertically or horizontally, at the re
moter extremity of the line whose length we wish to ascertain. Thus,
if there be a house, or other erection, built with bricks, of the usual
size ; then four courses in height are equal to a fooi^ and four in
length equal to a yard: and distances measured by means of these
will be tolerably accurate, if care be taken while observing the angle
subtended by the horizontal object, to stand directly in front of it.
A man, a carriagewheel, a window, a door, &c., at the remoter ex
tremity of the distance we \^ ish to ascertain, may serve for an ap
proximation. But in all cases where it is possible, let a foot, a yard,
or a sixfeet measure, be placed vertically, at one end of the line to
be measured, while the observer with his micrometer stands at the
other. Then, if h be the height of the object,
either  A x cot \ angle subtended
or A X cot angle
will give the distance, according as the eye of the observer is hori
zontally opposite to the middle^ or to one extremity of the object
whose angle is taken.
When a table of natural tangents is not at hand, a very near ap
proximation for all angles less than half a degree^ and a tolerably
near one up to angles of a degree^ will be furnished by the following
rules.
If the distant object whose angle is taken be 1 foot in length,
then
3437*73 6 the angle in minutes \ will give the distance in
or 206264 ^ the angle in seconds i feet.
If the remote object be 3, 6, 9, &c., feet in length, multiply
the former result by 3, 6, 9, &c., respectively.
Ex. 1. What is the distance of a man 6 feet high, when he sub
tends an angle of 30 seconds ?
• ForPD: DA:: AB: BR = ^^ ; andOD:OP::BR!RQ =
A B' . O P
O D . D P*
CHAP, v.] HBI0HT8 AND DISTANCES. 161
206264 X 6 r 30 = 206264 r 5 = 412528 feet =t= 137509
yards, the distance required.
Ex, 2. In order to ascertain the length of a street, I pat up a foot
measure at one end of it, and standing at the other found that mea
sure to subtend an angle of 2 minutes : required the length of the
street.
343773 r 2 = 171886 feet = 57295 yards.
3. By means of the velocity of sound. — Let a gun be fired at the
remoter extremity of the required distance, and obserre, by means of
a chronometer that measures tenths of seconds, the interval that
elapses between the flash and the report : then estimate the distance
for one second by the following rule, and multiply that distance by
the obserred int^val of time ; the product will give the whole dis
tance required.
At the temperature of freezing, 32% the Telocity of sound is 1100
feet per second ; for lower temperatures deduct, or for higher tem
peratures add, half a foot per second for every degree of difference
from 32^ on Fahr. thermometer ; the result will show the velocity of
sound, very nearly, at all such temperatures.
Thus, at the temperature of 50% the velocity of sound is,
1100 X i (50 — 32) = 1109 feet.
At temperature 60% it is 1100 + i (60 — 32) = 1114 feet.
For a more accurate method of determining the velocity of sound,
I page 281.
162 CONIC SECTIONS. [part I.
CHAP. VI.
CONIC SECTIONS.
Sect. I. Dejiniti&ns*
1 . Conic /Sections are the figares made by a plane cutting a 6one.
2. According to the different positions of the cutting plane there
arise five different figures or sections, viz., a triangUy a eirde^ an
ellipse^ an hyperbola, and a parabola: of vt^hich the three last aie
peculiarly called Conic Sections,
3. If the cutting plaoe pass throngh the vertex of the cone,
and any part of the base, the section will be a iHan^y as AB.
(Fig. 115.)
4. If the plane cut the cone parallel to the base, or make no angle
with it, the section will be a circle , as A B. (Fig. 116.)
5. The section is an ellipse when the cone is cut obliqnely through
both sides, or when the plane is inclined to the base in a less angle
than the side of the cone is, as A B C D. (Fig. 120.)
6. The section is a parabola when the cone is cat by a plane
parallel to the side, or when the cutting plane and the side of the
cone make equal angles with the base, as P^ A P^ . (Fig. 138.)
7. The section is an hyperbola when the cutting plane makes a
greater angle with the base than the side of the cone makes, as
PjAP,. (Fig. 130.)
iVb/e.— In all the above definitions the cone is supposed to be a
right cone.
8. If all the sides of the cone be continued through the vertex,
forming an opposite equal cone, and the plane be also continued to
cut the opposite cone, this latter section is called the opposite hyper^
bola to the former, as S^ B S, . (Fig. 130.)
9. And if there be two other cones, with their axes in the same
plane and their sides touching the sides of the former cones, then
will the same plane cut all the cones and form four hyperbolas, as
P,AP«, R, CRo, S^BS^, Qj DQ^ (fig. 118), each opposite pair
of which are similar ; these hyperbolas are said to be eonjtigate to
each other, and the figure thus formed is called the figure of ike
conjugaJte hyperbdas, as fig. 118.
^(rfe.— In the following definitions, the letters refer to fig. 117
for the ellipse; fig. 118 for the hyperbola; and fig. 119 for the
parabola.
CHAP. VI.] CONIO 8I0TI0N8. 103
10. The veriice9 of any section are the points where the cutting
pkne meets the opposite sides of the cone, as A and B.
Cor, Hence the ellipse and the opposite hyperbolas hare each two
vertices; but the parabola only one; unless we consider the other as
at an infinite distance.
1 1. The mqfor amt^ or iransverte diameter of a conic section, is
the line or distance A B between the Yertioes.
Cor. Hence the axis of a parabola is infinite in length, A B being
ody a pari of it.
1 2. The eenhre O is the middle of the axis.
Cor, Hence the centre of a parabola is infinitely distant from the
vertex. And of an ellipse, the axis and centre lie within the curve :
bot of an hyperbola, without.
Id. The mtHor^ or conjugate aariSj is the line C D, drawn through
the centre perpendicular to the transverse axis, and bounded each
Way by the curre.
CJcr, Hence the parabola has not any conjnsate axis, unless we
aappoee it at an infinite distance, and infinite in length.
14. A Umgemi to a curve at any point, is a line as A^ T^ , which
tooehes the carve in that point; but being produced either way docs
not cut it.
15. A diameter is any right line, as A^Bj, drawn through the
csentre, and terminated on each side by the curve ; and the extremi
ties of the diameter, or its intersections with the curve, as A ^ and
H J , are its vertices.
Cor, Hence all the diameters of a parabola are parallel to the
Axia, and infinite in length. Hence, also, every diameter of the
ellipse and hjrperbola has two vertices; but of the parabola, only one;
aaleaa we consider the other as at an infinite distance.
1^. The coi^ugaU to any diameter is the line drawn throngh the
centre, and pandlel to the tansent of the curve at the vertex of the
diameitf. So, 0, D, , panllel to the tangent at A ,, is the conjugate
toAjBji.
17. An mrdimate to aay diameter is a line parallel to its conjugate,
or to tlie tangent at its vertex, and terminated by the diameter and
carve. So^ K I and K, Aj are erdinates to the axis A B ; and K, I
•ad fi I ordiaates to the diameter A, B^ .
Cor. ilesoe the ordinatea of the axes are perpendicular to it; but
of other diameters, the ordinates are oblique to them.
18. An oiirisfa is a pvt of any diameter, contained between its
vertex and an ordinate to it; as AK or BK, and A, E, or B, S, .
Cor. Hence, in the ellipse and hyperbola, every ordinate has two
abec ioom ; \ml in the parabola only one; the other vertex of the
diaeieter beiag infinitely distant.
19. The ^wrtex from which the abcissa are measured is called the
origin of those abcissse, and any abcissa and its ordinate are called
co^frdinaieSi as AK and KI.
20. The parameter of any diameter is that double ordinate which
is a third proportional to the transverse and conjugate axes in the
M 2
164 CONIC 8BCTI0N8. [PART I.
ellipse or hyperbola, and to any abscissa and its ordinate in the para
bola. The parameter of the transyerse axis is called the principal para
meter^ or the lattis rectum; thus, if A^ B. : C, D, :: Cj D, : I^ I^,
then Ig I4 is i\\e parameter of A ^ Bj ; and if AB : CD :: CD : I'j I3,
then I J I3 is the lotus rectum,
21. The point where the parameter cuts the transyerse axis is
called \he focuB^ as E, F; and the distance of the focus from the
nearest yertex of the same axis is called the focal digtance^ as AE, BF.
Cor, Hence, the elHpse and hyperbola haye each two foci, but the
parabola only one. The foci (or burning points) were so called
because all rays are united or reflected into one of them, which pro
ceed from the other focus, and are reflected from the curye.
22. The point £ is also frequently called the poie; and a line
drawn from any point in the curye to the pole, is called the radius
vector y as E I. And the angle B £ I, contained between the radius
yector and the transyerse axis, is called the traced angle,
23. The directrix is a right line drawn perpendicular to the trans
yerse axis of a conic section, through an assignable point in the
prolongation of that axis; such that lines drawn from any points in
the curye parallel to the axis to meet the directrix, shall be to lines
drawn from the same points to the focus, in a constant ratio for the
same curye.
Thus, if A E : A T :: £ I^ : I«y :: E I3 : I3 « :: E I : I X, then
X Y is the directrix.
In the ellipse A £ is less than A T.
In the parabola A E is equal to A T.
In the hyperbola A E is greater than A T.
24. The subtangent to any point in a curye, is that portion of the
transyerse axis which is contained between the tangent and ordinate
to the same point, as K^ T, .
25. A normal at any point is a line drawn from that point per
pendicular to the tangent, to meet the transyerse axis, as A ^ L.
20. The Radius q/* Curvature of a conic section or other curye, is
the radius of that circle which is precisely of the same cunratnre as
the curye itself, at any assigned point, or the radius of the circle
which fits the curye and coincides with it, at a small distance on
each side of the point of contact. The circle itself is called the oseu
latorg circle, or the eguicurve circle ; and if the curye be of inces
santly yarying curyature, each point has a distinct eqaicarye circle,
the radius of which is perpendicular to the tangent at the point of
contact.
27> An asymptote is a right line towards which a certain cunre
line approaches continually nearer and nearer, yet so as never to
meet, except both be produced indefinitely. The hjrperbola has two •
asymptotes, as UV, WZ.
CHir. VI.] CONIC SBCTIONS: ELLIPSE, 165
Sbct. II. Properties of the Ellipse.
1. Let the frostum of a right cone AGBH (fig. 120), he cut by
apkne fonning a tangent to each end, then will the section (ACBD)
thus prodaced be an ellipse. Let dszQBy the diameter of the lesser
end; D = AH the diameter of the greater; A = AG, or BH, the
dant height; t=iAB^ the transverse diameter of the ellipse ; c = C D,
the eonjugile diameter; andysEF, the distance between the two
foci; then,
^=A2 + Drf (1.)
c» = D^ (2.)
/ =A (3.)
1 Lety=IK (fig. 117), any ordinate; a;=:AK, its abscissa
nettored from A ; ;y =: O K, its abscissa measured from the centre,
O;0 = IE, the radiup vector from the focus £; i9= the traced
»a^ IBF; and ^= >^i^'— i<^; then,
y = f ^(/4r««) (4.)
viiich b the equation to the ellipse when the absciss® are measured
ftm the vertex ;
y=f Va<'**) (5.)
vbich is the equation when the abscissas are measured from the
•tttre; and
^^ it — g.coBS ^ '^
thich is called the polar equation^ and finds its principal use in the
miei^mtions of astronomy.
3. For the relation between the abscissse and ordinates in any
•%ie, we have.
As the square of the transverse azis^
: the square of the conjugate ;
: : the rectangle of the abscissas,
: the square of their ordinate.
Or, /«:tf^::«(/«):y» (7.)
"Rw sune proportion obtains between the ordinate and abscissae to
•^ditmetcr; or putting «, for Aj^j (fig. 117); w^ for B^Kj,
^^y,forIKi; ^^^
t,*:e,^::w,x^:y,* (8.)
^ tarn of the squares of any pair of conjugate diameters is
^ to the sum of the squares of the two axes; or putting t^ for
^i^ (fig. 117), and e^ for C^ D^ ; then
t^ + e'^t^^^e,^ (9.)
^U the parallelograms that can be circumscribed about an ellipse
•reequil to the rectangle of the two axes; or = ct (10.)
106 CONIO 8BCTI0N8: BLL1P8B. [pABT I.
The sum of the two radius vectors, drawn from the two foci, to
any point in the curve, are equal to the transverse axis; or putting
e = £ I (fig. 117), the radius vector from £, and V = F I, the same
from F ; we have
V hr = / (11.)
Cor. The distance of either focus from either extremiljr of the
conjugate diameter is ^oal to half the tmnsverae axis ; or,
CE =CF =  (12.)
The square of the distance between the two foci equals the differ
ence of the squares of the two axes ; or,
r^t'^e' (1«.)
4. tf TM (fig. 121) be the tangent to an ellipse at any point I,
and let T be the point in which the tangent meets the transverse
axis produced; also from the two foci draw FM, EN, perpendicular
to the tangent, and produce E N and F I to meet each other in P ;
then,
OK : AO :: AO : OT; or, OT —  (14.)
Cor. If there be any number of ellipses described on the same
transverse axis (fig. 122), and any ordinate be drawn, cutting all the
ellipses in the points I^, I^, I^, I^, &c., the tangents to the several
ellipses at those points, wilt all meet the transverse axis produced in
one common point, T 0^')
The angles made by tiie two radius vectors and the tangent arc
equal; or,
2LTIE=:/.LIF (16.)
The points M and N (fig. 121) fall in the circumference of a circle
Ivhose diameter is the transverse axis, AB (17.)
Also, EN. FMarCO'tfc ^1 J (18.)
And, EN*
= (I)'t (•••)
If m be in the middle of K I , , then, A m prodctced wOi meet the
two tangents TQ, BQ, in their point of intersection, Q ... (20.)
If the ordinate 1 1 ^ passes through the focus, tben the point T of
intersection of the tangent and the transverse axis prodac^ will be
a point in the directrix (21*)
c'
In any ellipse the parameter = — (^^0
5. Let R ss the radius of curvature at any point in an ellipse, and
V, «, the distances of the same point from the two foci; then
R = *^ (23>
CBAf. YI.] COMIC MCTIOVS: BLLIP81. 167
It is A mtunmum at the extremity of the conjugate axis, when
*^f. ("•>
And a minimum at the extremity of the transverse axis, when
»* = ^ (25.)
The area of any ellipse is a mean proportional hetween the areas
of the circles deambed on its two axes (^^0
P90BLSM8 jaiLATINO TO THX ELLIPSE.
^«OB. I. To find the two foci E^ Fy of an ellipse^ of which the
transverse cune A B, and conjugaU C D, are given.
Prom C or D AS a centre, with a radius equal to the semi trans
^'cvce axis AC or BO, describe an arc cutting the transverse axis in
^ J, wLich are the two foci required. (Fig. 123.)
FiOB. II. To von^mct an ellipse whose two axes are given,
Knd the distance £F, from Prob. I., and let a fine thread, £IF£,
?* length = £F + A B, be put round two pins fixed at the points
^ ^; theo, if a pencil be put within the cord, and carried round,
^^>^ being taken diat the cord is alwavs tight, the pencil will dc
•*^*"ibe an ellipse, A I C B D A. (Fig. 1 24*)
^^^^41. III. The transverse and conjugate amis of an ellipse being
given^ to describe the same with a trammel,
let the distance between the pencil A and the first pin B, be
^^^ to half the conjugate axis, and the distance between A and
^^^ leeond pin C, be equal to half the transverse ; then, the pins
****iBg pot into the grooves of the trammel, and the pencil A being
**^^>^ will describe the ellipse. (Fig. 125.)
OQierwise, — ^Let there be provided three mlers, of which the two
^^) FH, are of the same length as the transverse axis AB, and the
^^vdHP, equal in length to £F, the focal distance. Then, con
^^^cting these rulers so as to move freely about £ F H and I^ their
^•iimtiun I will always be in the curve of the ellipse : so that,
^f tkere be alits mnning along the two rulers, and the apparatus
^*ned fiteiy aboat the foci^ a pencil put through the slits at their
^^ sf intenectaon will deacribe the carve. (Fig. 126.)
Pbob. ly. To find the two axes of any given ellipse,
^W tny two pamUel lines across the ellipse, as M L, F K : bisect
"•"» in the points I and D, through which draw the right line
168 CONIC sections: hypbrbola. [part i.
N I E P, and bisect it in O. From O as a centre, with any adequate
radius, describe an arc of a circle cutting the ellipse in the points
G, H. Join O, H, and parallel to the line G, H, draw through O
the minor axis CD; perpendicular to which through O draw AB,
which will be the major axis. (Fig. 127.)
pROB. V. From any given point out of an ellipse to draw a
tangent to it.
Let T be the given point, through it and the centre C draw the
diameter A B ; and parallel to it any line H I terminated by the curve.
Bisect HI in O ; and C O produced will be the conjugate to A B.
Draw any line TS = TB, and make TR = TC. Draw RA, and
parallel to it, SP cutting AB in P. Through P, draw PM parallel
to CD, and join TM, which will be the tangent required. (Fig. 128.)
Prob. VI. To find the length of an elliptic arclj C I^ .
Produce the height CE to O, the centre of the ellipse. Join
I J and O, and from O as a centre with radius C O eqttai the semi
conjugate, describe an arc cutting I , O in a. Bisect 1 1 a in 5, and
from O as a centre with radius O b describe the arc bcy catting C O
produced in c; then the circular arc ^c is equal in length to half the
elliptic arc I, C Ig . (Fig, 129.)
Sect. III. Properties of the Hyperbola.
1. Let two opposite right cones (Fig. 130) be cut by a plane
making a less angle with their axes than the sides of the cones make,
then the sections thus produced will be two opposite hyperbolas. Let
d = AHy the lesser diameter of the cone at the vertex of one hyper
bola ; D = B G, the greater diameter of the cone at the vertex of the
opposite hyperbola ; /t = A G, or B H, the slant height ; / = A B, the
transverse axis; c = CD, the conjugate axis; and /*= £F, the dis^
tance between the two foci ; then we have
/«=:A2Drf . . . (1.)
c' = Drf . . . . (2.)
f = h . . . . (3.)
which will be seen to be identical with the corresponding equations
for the ellipse, with the exception of the first, which, however, only
differs in the sign of the last term. This at once indicates a generu
analogy between the properties of the two curves ; and if we employ
the same letters to represent the corresponding lines in figures llTi
121, 122, and figures 118, 131, 132, the formula already given for
the ellipse will apply to the hyperbola, only making the following
alterations.
CHAP. VI.] 4x>Nic sections: hypkbbola. 169
(4.) Becomes y = T/V/ ^'* "'' *'^'
(5) « 9=fAji''\n
(6.) „ f> = ^
I < + ^ • COS /3
(7.) and (8.) remain unaltered.
(9.) Becomes <,* — c^' = f* — c*.
(10.) All the parallelograms that can be inscribed between four
conjugate bjrperbolaSy are equal to the rectangle of the two axes ; or,
= et
(11.) Becomes V— © = ^
(12.) „ OEorOF = ACorCB.
(13.) „ r^f^e.
(14.) to (25.), both inclusiTe, remain unaltered.
2. Besides these, however, there are several curious properties
which relate to the tuymptotes of the hjrperbola. Let 8), '3, 829
(fig. 133) be an hyperbola; OV and OZ its asymptotes; 9^ K«^,
a double ordinate to the transverse axis; O^r^, Oz^^ Oz^^ Oz^^ Oj»^,
abscissa taken on the asymptote OZ; and;!;, 'i>^8 '^^ ^3 'si ^4 '49
z^ «5, ordinates to the same, drawn parallel to the other asymptote
OV:
Then the parallelograms 0©^ *, ^r,, Ov^ 9^ z^y O©,, «, z^y &c.
ire all equal ; or
0*1 ^1 *i = ^^« • *2 *s =0^3 .^3 *j, =04r4 .«4«4 (26.)
/«
Also, m#i . <i *a = ««5 . «i 85 = ^*s' = J • • (27.)
And the triangle O T I, (Fig. 131) = the triangle O B Q . (28.)
Also, if the abscissae Oz^^Oz^^Oz^^ &c. of any h^rperbola, be
taken on one of the asymptotes in an increasing eeometncal progres
sion, the ordinates z^9^y z^ ^^ , z^ 9^^ &c. parallel to the other asymp
tote are in decreasing geometrical progression, having the same
imtio (29.)
And, when the distances Oz^y Oz^^ &c. are in geometrical pro
gression, the asymptotic spaces z^ 9. So z^^ Zy «. 9^ jT,, &c. will be
m arithmetical progression, and will, therefore, be analogous to the
logarithms of the former (30.)
The radix of the system of logarithms will depend upon the
value of the angle made by the two asymptotes. In Napier 9
logarithms V O Z is a right angle ; while in the common logarithms
it is 24** 44' 254" ♦.
* See Sect. 6, page 93.
170 COXIC 8BCTI0K8: HYFIRBOLA. [PABT I.
3. In the cmse of foar conjngate bypcrboIa8, the transyerse and con
jugate axes of one pair become the conjugate and transTerse of the
other pair, and th^efore they are said to be matually conjugate to
each other.
PBOBLRMS RELATING TO HYPERBOLAS.
Prob. I. To detcrihe an Hyperbola of tckich the tranwerM axis and
twofod (ire given.
Let one end of a long ruler, £0 (fig. 134) be fictttened at one of
the foci, £, so as to turn freely about that point as a centre. Then
take a thread F I , O, of such a length, that when one end is fixed at
the other focus F, and the other end to the ruler at O, the doubled
part will just reach to B, one end of the transverse axis. Then if the
ruler E O be turned about the fixed point E, at the same time keep
ing the thread O I , F always tight, and its paK I , O close to the side
of tlie ruler, by means of the pencil I j ; the curve line B S . de
scribed by the motion of the pencil I, is one part of an hyperbola.
And if the ruler be turned, and move on the other side of the fixed
point F, the other part B S^ of the same hyperbola may be described
after the same manner. But if the end of the ruler be fixed in F,
and that of the thread in £, the opposite hjrperbola P ^ A P, jao^y be
described.
Othencise; also by continued motion. Let E and F (Fig. 135)
be the two foci, and A and B the two vertices of the hyperbola.
Take three rulers, C D, D I, I F, and fix them so that E D s: G F ax
A B, and DC =: EF; the rulers D I and 6 1 being of an indefinite
length beyond I, and having slits in them for a pin to move in ; and
the rulers having holes in them at £ and F, to fiasten them to the foci
£ and F by means of pins, and at the points D and C they are to be
joined by the ruler D C. Then, if a pencil be put in the slits, at the
common intersection of the rulers D I and F I, and moved along,
causing the two rulers F I, ID, to turn about the foci E and F, that
pencil will describe the portion A P j of an hyperbola.
Prob. II. To describe an hyperbola of which the two axes aregiveuy by
finding points in the curve.
Let A B be tlie transverse axis, and C D the conjugate. First from
O as a centre, with radius equal A C or C B, describe arcs cutting
the tuansverse axis produced in £ and F, which will be the two foci.
Then assume any number of points, a^t a,, a^y &c. beyond F, and
from £ as a centre, with radii equal to Ba,, Bao, Ba,, &c., describe
arcs ^,9 ^^9 ^89 ^^* ' ^^^ ^^^^ F as a centre, with radii equal to
Aa,, Aa.„ Aa^, &c., describe arcs cutting the former arcs in the
points c^^c^^ C3, &c., which will all be points in the hyperbola re
quired. (Fig. 136.)
CHAP. VI.] CONIC BJraTlONB: PABABOLA. 171
Prob. III. To deteriht an kyperhokL, of which the transvene axis A B,
and any abscissa A O, and its doubU ordinate H I, are given.
Through A draw a h parallel to H I ; from H and I draw H a and
I h parallel to A O ; then divide aH and &I, each into any number of
equal parta, 1, 2, 3, &c., and through the points of division draw lines
radiating to A as a centre ; also divide K H and K I into the same
number of equal parts, and through the points of division draw lines
nuliatiog to B, then will the several points «,, Cg, C3, &c, in which
these lines intersect the former lines, be so many points in the
hyperbola required. (Fig. 137.)
Sbct. IV. Properties of the Parabola.
1. If the right cone KHI (fig. 138) be cut by a plane parallel
to the side KH, then will the section P^ A P,, thus produced, be a
parabola. Let the cone also be cut by a plane, M A, perpendicular to
the plane ^ the parabola, and by another plane, G A, jiarailel to the
base of the cone, and both passing through the point A ; bisect the
distance M O (between their two vertices) in L, and through L, take
a plane L T, parallel to the base of the cone, and let S be the focus
of the parabola. Let & = B P^, half the base of the parabola ; d =
OA, the dinneter of the cone at the vertex of the paral)ola ; /= A E,
ike focal diBtanoe ; and A s K A, the alant height of the cone above
dieaame; iImo
The Mm TX, in which the plane L T meets the plane of the para
bola, is its directrix ... ... (1.)
d^
TA=/=lj (2.)
b
2
fVd w
2 (M O) = 4/ = the parameter . . (4.)
2. Let /> = I^ I3 (fig. 119), the parameter of a parabola;
« vs A &, aay aheeisaa ; jf bb I, X, the corresponding ordinate ; v =
£ I, the radiiB vector; and /9 ss the tnced angle II B, £ being the
focus; then
f^px. . . . (5.)
f> = , ^^ ^ . . (6.)
1 q: cos g ^ '
the ayiitiopg to the paiabok : im the latter of which, or the polar
eauation, the sign + obtains when K is between A and S, and —
when K is below £.
(4^ f p\\
Rad. of curvature at I = „ ^^^ . (7.)
2 >/;t> ^
172 CONIC sections: parabola. [part i.
At the vertex, A, x vanishes, and we have rad.
of curv. at vertex = i /> • • . (8.)
3. In the same figure, where XY is the directrix, the following
properties ohtain ; viz.
AE = AT, EIj = Igy, EI3 =130:, EI =IX, &c.(9.)
AsAK : AK, :: KI* : K.P 2; or ^^ = ^^ ^ v (10.)
where A K and A R, are any ahscissss, and K I, K3 P ^ their corre
sponding ordinates. Also
EI = AK + AE, EPj = AK3 4 AE . (11.)
AE = ni=Lil (12.)
Ij I3 heing the parameter = p,
As/?:K,P^ + Kgl :: K,Pj  K,I :AK3  AK, ^ . .
or, asjt? : P^a :: aP^ : KgK3 i
4. Again, let A , T ^ he the tangent to a parahola at any point A,
and let B ^ A J Y he drawn through A . parallel to the axis A B ; let
A J L he perpendicular to A^ T. ; then is K, T, the suhtangent, A . L
the normal, K^ L the suhnormaJ; and the following properties obtain;
viz.
angle EA, T^ = angle ET, A^ = angle T,Aj Y . . (15.)
angle LAjBj = angle LA^E (1^0
EA,= ETi (17.)
AKj = ATj (18.)
subtangent K, Tj = 2 AKg (19.)
V
subnonnal K, L = 2AE = ^a constant quantity . • (20.)
5. In figure 139 also, where CQ is a tangent to the parabola at the
point C, and IK, OM, QL, &c., parallel U) the axis AD.
Then IE : EK :: CK : KL . . (21.)
and a similar property obtains, whether CL be perpendicular or ob
lique to TD.
The external parts of the parallels IE, TA, ON, QL, &c., are
always proportional to the squares of the intercepted parts of the tan
gent; that is,
the external parts IE, TA, ON, QL, \
are proportional to CI', CT', CO% CQ^ [ . . (22.)
or to the squares CK^ CD^, CMV CL', '
CHAP. VI.] CONIC sections: pababola. 173
And as this property is common to every position of the tangent,
if the lines IE, T A, ON, &c., he appended to the points I, T, O, &c.,
of the tangent, and movahle ahout them, so as always to hang verti
cally, and of such lengths that their extremities E, A, N, &c., he in
the curve of a parahola in any one position of the tangent ; then
making the tangent revolve ahout the point C, the extremities E, A, N,
&c., will always form the curve of some parahola, in every position
of the tangent.
The same properties, too, that have heen shown of the axis, and
its ahscisses and ordinates, &c., are true of those of any other dia
meter.
PROBLEMS RELATING TO THE PARABOLA.
Pbob. I. To anutruet a Parahola^ of which the base and height are
given.
Construct an isosceles triangle ABD (fig. 140) whose hase AB
shall he the same as that of the proposed parahola, and its altitude
CD twice the altitude G V of the parahola. Divide each side AD,
DB, into 10, 12, 16, or 20, equal parts [16 is a good numher, hecause
it can he obtained by continual bisections], and suppose them num
bered 1, 2, 3, &c. from A to D, and 1, 2, 3, &c. from D to B. Then
draw right lines I, 1; 2, 2; 3, 3; 4, 4; &c., and their mutual inter
section will beautifully approximate to the curve of the parahola
AVB.
Pbob. II. To describe a Parabola by finding points in the curve^ an ab
scissa A B, and its double ordinate C D, being given.
Through A (fig. 141) draw a b parallel to CD ; and through C and
D draw aC, 6D parallel to AB; then divide aC and bD into any
convenient number of parts, 1,2, 3, &c., and through these divisions
draw lines radiating to A ; also divide B C and B D into the same
number of parts, and through the divisions draw lines perpendicular
to CD, then the points c,, c^* c^, &c., in which these lines intersect
the former, will be in the parahola.
Let the ruler, or directrix BC (fig. 142) be laid upon a plane with
the square ODO, in such a manner that one of its sides DO lies along
the edge of that ruler; and if the thread F M O, equal in length to
D O, (Uie other side of the square,) have one end fixed in the extre
mity of the ruler at O, and the other end in the focus F : then slide
the side of the square D O along the ruler B C, and at the same time
keep the thread continually tight by means of the pencil M, with its
part MO close to the side of the square DO; so shall the curve
A M X, which the pencil describes by this motion, be one part of a
parabola.
And if the square be turned over, and moved on the other side of
the fixed point F, the other part of the Rame parabola A M Z will be
described.
174 CONIC 8BCTI0N8 : PARABOLA. [PART I.
Pbob. III. Any right line being given in a parabola^ to find the corre
sponding diameter; aleoy the axis, parameter^ and focus.
Draw HI (fig. 143) parallel to the given line DE. Bisect D £, and
H I, in # and O, through which draw A O O for the diameter. Draw
H R perpendicular to A O and hisect it in B; and draw V B parallel
to A O for the axis. Make V B : H B : : H B : parameter of the
axis. Then ^ the parameter set from V to F gives the focus.
Prob. IV. To draw a tangent to a Parabola,
If the point of contact C (fig. 144) he given, draw the ordinate,
CB, and produce the axis until AT = A B : then join T C, which will
he the tangent.
Or if the point be given in the axis produced: take A B ^ AT,
and draw the ordinate B C, which will give C the point of contact ;
to which draw the line T C as before.
If D be any other point, neither in the cunre nor in the axis pro
duced, through which the tangent is to pass: draw DE O perpendi
cular to the axis, and take D H a mean proportional between DE and
DO, and draw H C parallel to the axis; so shall G be the point of
contact, through which and the given point D the taD*mit D C T is to
be drawn.
When the tangent is to make a given angle with the ordinate at tbe
point of contact : take the absciss A I equal to half the parameter, or
to double the focal distance, and draw the ordinate I £ : also draw
A H to make with A I the angle A H I equal to the given angle; then
draw H C parallel to the axis, and it will cut the curve in C the point
of contact, where a line drawn to make the given angle with C B will
be the tangent required.
CHAF. VII.] CURVES. 175
CHAP. VII.
PROPBBTIER OP CURVES.
Sect. I. Definitions. #
1. A curve line^ as already defined, (Def. 5, page 98,) ia a line
which continually changes its direction.
2. Apiane carve, is one the several points of which are all situate in
the Mme plane ; and a curve of double curvature is one whose several
points are not aJl in the same plane, and which, therefore, curves in
two directions.
3. Curve lines have heen further divided into AlgebraiccU or Geo
metrieat^ and TrtxMtemiental or MeckanicaL
4. An tdgehrauud curve, is one in which the relation which the
abseista bears to the ordinate, taken for any possible point in that
corye^ may always be expressed by a constant algebraical formula.
5. A transcendental cuTvCy is one in which no such constant rela
tion between its abscissse and ordinates exists, and which can, there
fore, only be described by mechanical means.
6. The algebraical formula which expresses this relation is called
the equation of the curve; and curves are classed into orders, accord
ing to the number of dimensions contained in such equations. Thus
any curve line which might be represented by the equation
a: a + ^x H cy + da^ + exy +/y*,
would be called a line of the second order, because the abscissa {ai)
and ordinate (y) are involved only to the second power; and a curve
which might be expressed by
= • + 6« + cy h <^^ h ««y +ff h gas" + hwy + hf» + If
would be called a line of the third order, because the same quantities
are involved to the third power. The letters a, &, c, &c., are merely
the coefficients depending upon the particular curve expressed by the
equation.
7. An am/mpMe to a curve is a straight line, to which the curve
line continually approaches, without ever meeting it; as the asymp*
iotes U V and W Z, to the hvperbola, fig. 118.
8. If a curve cuts itself by passing twice through the same point,
as A, fig. 145, this point is called &punctum duplea: if three times, as
K, it is called a punctum tripUx^ &c.
170 curves: conchoid. [part i.
9. The oval contained between A and C, fig, 145, is called a
nodus: when the distance between A and C becomes indefinitely
small, so that the nodus vanishes, the point A is called a punclum
cofijugatum,
1 0. A point in which two branches of a curve terminate, is called
a cuspia,
1 1 . The involute of any curve, is another curve traced by the end
of a string in being unwound from the first curve, which is called the
evolute. Thus, let A B C (fig. 1 46) be a curve having a thread laying
in contact with every part of the same ; then, if the thread be un
wound from A, the curve line A D £« which its end will describe,
will be the involute to the curve ABC, which latter will be the evo
lute of the <?tirve A D £.
The principal lines connected with curves, such as tangenty ordinate^
absciBSoe^ &c., having been already defined in the first section of
Chap. VI., it is unnecessary to recapitulate them here.
12. The equations of the conic sections have been already given,
for the ellipse at page 165, the hyperbola at page 169, and the para
bola at page 171 : the equation for the circle is
y = s/dx — aj*,
d being the diameter ; the whole of these four curves are therefore
lines of the second order, the quantities a; and y, only being raised to
that power. We shall now pass on to describe the principal pro
perties of those curves which are most frequently employed for useful
purposes.
Sect. II. The Conchoid.
Conchoid^ or Conchiles^ (from the Latin concha^ a shell,) is the
name given to a curve by its inventor, Nicomedes^ about 200 years
before the Christian era.
The conchoid is thus constructed: AP and BD (fig. 147) being
two lines intersecting at right angles: from P draw a number of
other lines PFDE, &c., on which make always DE = DF = AB
or BC; so shall the curve line drawn through all the points E, £, E,
be the first conchoid, or that of Nicomedes; and the curve drawn
through all the other points, F, F, F, is called the second conchoid;
though, in reality, they are both but parts of the same curve, having
the same pole P, and four infinite legs, to which the line DBD
(called the directrix) is a common asymptote.
The inventor, Nicomedes, contrived an instrument for describing
his conchoid by a mechanical motion, in the following manner: in the
ruler DD (fig. 148) is a channel or groove cut, so that a smooth nail
firmly fixed in the movable ruler CA, in the point D, may slide
freely within it: into the ruler AP is fixed another nail at P, for the
movable ruler AP to slide upon. If, therefore, the ruler A P be so
moved as that the nail D passes along the groove D D, the style, or
point in A, will describe the first conchoid.
CHAP. VII.] 0URVB8 : CI880ID. 1 77
CoDchoids of all possible Turieties may also be constructed with
great facility by Mr. Jopling's apparatus for curves, now well known.
1. Let AB = BC == DE = DF = a, PB = ft, BO = EH = x,
and OE = BH = y : then the equation to the first conchoid will be
«* (ft I «)* + xy = a (ft + «)*,
or, a?* + 2ftd^ + ft«a?' + x'f =^0^1^ + 2a'fta? + a^«*;
and, changing only the sign of x^ as being negative in the other
curve, the equation to the second conchoid will be
a?' (ft  xY h a?V = a^ (*  x)\
or, «* — Zhx" + h^x^ + x^'f = a" ft — 2aftx + aV.
2. Of the conchoids expressed by these two cauations (or rather
the same equation with different signs), there are three species ; first,
when a is lets than ft, the conchoid will be of the form shown in
fig. 148, the inferior branch having a punctum conjugatum at A ;
secondly, when a is equal to ft, the conchoid will assume the form
shown in fig. 1 49, the punctum conjugatum giving place to a cuspU at
B ; and thirdly, when a is greater than ft, the cuspis becomes a noduSj
the conchoid taking the form shown in fig. 150.
Newton approved of the use of the conchoid for trisecting angles,
or finding two mean proportionals, or for constructing other solid
problems. But the principal modem use of this curve, and of the
apparatus by which it is constructed, is to sketch the contour of
the section that shall represent the diminution of columns in ar
chitecture.
Sbct. III. The Ciuoid or Cyewid.
The eisMoid is a curve invented by an ancient Greek geometer and
engineer named Diodes^ for the purpose of finding two continued
mean proportionals between two given lines. This curve admits of
an easy mechanical construction ; and is described very beautifully
by means of Mr. Jopling's apparatus.
At the extremity B of the diameter A B (fig. 151), of a given circle
AOBo, erect the indefinite perpendicular eBE, and from the other
extremity A draw any number of right lines, AC, AD, AE, &c.,
catting the circle in the points R, O, M, &c. ; then, if C L be taken
= AR, DO = AO, EN = AM, &c., the curve passing through the
points A, L, O, N, &c. will be the eissoid.
1. Here the circle AOBo is called the generating circle; and AB
is the axis of the curves ALON, &c., A/ow, &c., which form a
cuspii at A, and, passing through the middle points O, o, of the two
semicircles, tend continually towards the directrix eBE, which is
their common asjrmptote.
2. Letting fall perpendiculars LP, RQ, from any corresponding
points L, R ; then is A P = BQ, and A L = CR.
178 curves: cycloid. [pabt i.
3. Also, AP : PB :: PL^ : AP*. So that, if the diameter AB of
the circle = a, the abscissa A P = «, and the ordinate P L = y ; then
« : a — jc :: y : jr, or ar* = (a — *)y%
which is the equation to the curve.
4. The arc AM of the circle = arc BR, and arc Ant = Br.
5. The whole infinitely long cissoidal space, contained between
the asjrraptote ^ B E and the curves NO LA, &c., and A Lou, &c., is
equal to three times the area of the generating circle AOBo.
Sect. IV. The Cycioidy and Epicycloid,
The cycloid^ or trochoid^ is an elegant mechanical carre first
noticed by Descartes^ and an account of it was published by Jffr
senne, in 1615. It is, in fact, the curve described by a nail in the
rim of a carriagewheel while it makes one revolution on a flat
horizontal plane.
1. Thu8,if a circle EPF (fig. 152),keeping always in the same plane,
be made to roll along the right line A B, until a fixed point P, in its
circumference, which at first touched the line at A, touches it again
at B after a complete revolution ; the curve APVPB described by
the motion of the point P is called a cychid.
2. The circle EPF is called the yeneraiing circle ; and the ri^t
line A B, on which it revolves, is called the base of the cycloid.
Also, the right line, or diameter CV, of the circle, which bisects
the base A B at right angles, is the axis of the cycloid ; and the point
V, where it meets the curve, is the vertex of the cycloid.
3 If P rfig. 153) be a point in the fixed diameter AF produced,
and the circle A E F be made to roll along the line A B as before, so
that the point A, which first touched it at one extremity, shall touch
it again at B, the curve PVP, described by the point P, is called a
curtate cycloid.
4. And, if the point P (fig. 154) be anywhere in the unproduced
diameter A P, and the circle A EF be made to roll along AB from A
to B; the curve PVP, is, in that case, called the inflected or prdaie
cycloid.
The following are the chief properties of the common cycloid.
5. The circular arc VE (fig. 155) is equal to the line EG between
the circle and cycloid, parallel to A B.
6. The semicircumference VEC is equal to the semibase CB.
7. The arc VG equals twice the corresponding choinl VE.
8. The semicycloidal arc VGB equals twice the diameter VC.
9. The tangent TG is parallel to the chord VE.
10. The radius of curvature at V equals 2 C V.
11. The area of the cycloid AVBGA is triple that of the circle
CEV; and consequently that circle and the spaces VECBG,
V E' C A Q\ are equal to one another.
CHAP. VII.] curves: QUADRATRIX. 179
12. A body falls through any arc LK (fig. 156) of a cycloid re
Teraed, in the same time whether that arc be great or small ; that is,
from any point L, to the lowest point K, which is the vertex re
versed: and that time is to the time of falling perpendicularly through
the axis M K, as the semicircumference of a circle is to its diameter,
eras 3141593 : 2. And hence it follows that if a pendulum be
made to vibrate in the arc L K N of a cycloid, all the vibrations will
be performed in the same time.
1 3. The eoolvte of a cycloid is another equal cycloid, so that if
two equal semicycloids OP, OQ (fig. 156), be joined at O, so that
O M is equal to M K the diameter of the generating circle, and the
•tring of a pendulum hung up at O, having its length equal to O K or
the curve OP; then, by plying the string round the curve OP, to
whieh it is equal, if the ball be let go, it will describe, and vibrate
in the other cydoid PKQ; where OP equals QK and OQ equals
PK.
14. The cycloid is the curve of swiftest descent : or a heavy body
will fall from one given point to another, by the way of the arc of
% cycloid passing through those two points, in a less time than by
ma^ Olher route. Hence, this curve is at once interesting to men of
sciattat aod to practical mechanics.
15. If iIm generating circle, instead of rolling along a straight line
is made to ndl upon the circumference of another circle, the curve
described by aiij point in its circumference is called an epicycloid,
16. This curve teives.its importance in practical mechanics, from
its being the curve m«i frequently adopted for the teeth of wheels.
17. If the generaduf circle revolves upon the convex circnm
ferenee of the quiescent cMe, as in fig. 157, it is then called an
eMeriar epie^fdoid A B CD. B«t if it revolves on the concave circum
ference, as in ^g, 158, the curve h called an interior epicycloid.
18. The area of an epicycloid, either interior or exterior, may be
found from the following proportion: as AB:3ABhBC ::a6c:
abD i: the area of the generating circle : the area of the epicycloid
J>BEebD,
19. If the diameter of the generating circle is equal to half that
of the quiescent circle, the epicycloid becomes a straight line; which
circmnstance has been taken advantage of for converting a continued
circular motion into an alternating rectilinear motion, in the manner
explained in the article on ^* Select Mechanical Expedients," in the
Appendix*
Sbct. V. The Quadratrix.
The quadrcUrix is a species of curve by means of which the quad
rature of the circle and other curves is determined mechanically.
For the quadrature of the circle, curves of this class were invented
by Dinoitrates and TechimltauseUy and for that of the hyperbola by
Mr. Perks. We shall simply describe in this place the quadratrix of
N 2
180 curves: catenary. [part i.
Dinottraies; in order to show its use in the division of an arc or
angle.
To construct this quadratrix, divide the quadrantal arc AB (fig. 159)
into any number of equal parts, A&, hc^ cd^ dB; and the radius AC
into the same number of equal parts, A/^ /y, gh^ AC. Draw radii
C^ Cc, &c., to the points of division upon the arc; and let lines
/li pm, &c.y drawn perpendicularly to AC from the several points of
division upon it, meet the radii in /, m, n, &c., respectively. The
curve A/mnD that passes through the points of intersection I, fii,&c.
is the quadratrix of Dinostrates.
The figure A CD n ml A thus constructed may be cut out from a
thin plate of brass, horn, or pasteboard, and employed in the division
of a circular arc.
Thus, suppose the arc IL (fig. 160) or the angle IKListobe
divided into five equal parts. Apply the side AB of the quadra
trix upon I K, the point B corresponding with the angle K. Draw
a line along the curve AS, cutting RL in F. Remove the instru
ment, and from F let fall the perpendicular FE upon IK. Divide
EI into five equal parts, by Prob. VII. page 113, and through the
points of division draw CM, DN, &c. parallel to £F. Then, through
their intersections M, N, O, P, draw the lines KM, KN, KO, KP,
radiating from K, and they will divide the angle IKL into five equal
parts, as required.
Note 1 . — If, instead of dividing the arc into equal parts, it were
proposed to divide it into a certain number of parts having given
ratios to each other; it would only be necessary to divide EI into
parts having the given ratio, and proceed in other respects as above.
Note 2. — If the arc or angle to be divided exceed 90 degrees,
bisect it, divide that bisected arc or angle into the proposed number
of parts, and take two of them for one of the required divittona
of the whole arc.
Sect. VI. The Catenary.
The catenary is a mechanical curve, being that which is assumed ^fl
by a chain or cord of uniform substance and texture, when it is hung"^
upon two points or pins of suspension (whether those points be^^
in a horizontal plane or not), and left to adjust itself in equilibrio— *
in a vertical plane.
The catenarian curve was for a long while regarded as the proper
form for the chains of a suspension bridge, but a slight consideration,
of the above definition will show its inapplicability. For, since the^
chains of a suspension bridge ought never to be of uniform substance^
and have also in addition to their own weight to support that of the
roadway, which latter is very differently distributed to the former;
the form of the curve which the chains will assume is materially
modified, and approaches more nearly to that of a parabola*.
• See page 207*
CHAP. VII.] CUBVBS: CATENARY. 181
Let AB (fig. 161) be the points of suspension of such a cord,
AaC^B the cord itself when hanging at rest in a vertical position.
Then the two equal and symmetrical portions AaCy CbB, both ex
posed to the force of gravity upon every particle, balance each other
predaely at C. And, if one half, as C&B, were taken away, the
other half, AaC, would immediately adjust itself in the vertical
position under the point A were it not prevented. Suppose it to be
prevented by a force acting horizontally at C, and equal to the weight
of a portion of the cord or chain equal in length to CM; then is C M
the measure of the tension at the vertex of the curve; it is also
regarded as the parameter of the catenary. Whether the portion
AaC hang from A, or a shorter portion, as aC, hang from a, the
tension at C is evidently the same : for, in the latter case, the resist
ance of the pin at a, accomplishes the same as the tension of the line
at a when the whole AaC hangs from A*.
Let the line C M which measures the tension at the vertex = />,
let Cd (or the deflexion as it is termed) ^=Xy ad (or the semispan)
= y, Ca=4r, CD = A, AB^d, CaA = C&B = i/. Then,
l.^=^(hyp.l«g'±*±^!£l±^)
= ^(hyp. log ^±^) = ^(hyp. log t±£)
te/>M.logten(45*» f \%);
where M == 2*30255851, the number by which the common loe
■rithnis must be multiplied to obtain the hyperbolic logarithms f;
and 8 = the angle which the tangent to the curve at the point a
makes with the horizon.
2. If the angle 8 of suspension made between the tangent to the
curve at A or B, and the horizon be 45°; then d : I :: 1 : 11346.
3. When /= 2rf, then h = 7966 d, and 8 = TT** 3'.
4. When the angle S of suspension is 56"^ 2B\ then />, a^ y, and
4r, are as 1, 0*81, 11995, and 1*5089 respectively. In this case f,
the tension at the point of suspension, is a minimum with re
elect to jr.
5. Generally, y = — tan SM . log tan (i 8).
Or, logy = log tan 8 f log {log cot (i 8) — 10} + '^22157  10.
This last formula serving to compute an approximative result.
• ThiP may easily be determined experimenully, by letting the cofd hung
very freely over a pulley at C, and lengthening or shortening the portion there
suspended, until it keqM AaC, in its due position ; then is the portion so hang
ing beyond the pulley eqnnl in length to C M.
t ^ce page 93.
182 curves: catrnaby. [pabt l
6. The distance of the centre of gravity of the whole cane 9#,
from the vertex = J (a? + ^^ — />)•
7. The radius of curvature — ^= — : this at the ^eitoxii
rad. curv. = p,
8. When S and p are given ; then
jr = j9 . tan 8
t s j9 . sec 8
« = p . sec (8  I) = ^^ , —
cos s
y = J9M . log tan (45° + i 8),
9. When 8 and z^ or ^ /^ are given s then
/? = ;3r . cot 8
^ = JBT . cosec 8
X ^ z . oosec 8 . versin 8
y = M4f . cot8 . log tan (45° + ^8).
10. When 8 and y are given : then
y
^ M . log tan (45° f i 8)
M . cos 8 . log tan (45° + i 8)
y . tanS
^ '^ M . log tan (45° f ^8)
y . versin 8
*^ l( . cos 8 . log tan (45° + i s)
11. When X and y are given ; then
log tan (45° f j 8) _ y ^
sec 8 . versin 8 VLx*
^rom which 8 may he found hy an approximative process ^ alac^
X _ ^ _ « . sin 8
sec 8 — 1 ' " versin 8 ' versin 8
in all these ca^es t is determined in kn^ of chain or cord of wh^
the catenary is actually constructed.
12. To draw the caUnary mechanicaUy, — If the distance 10
(fig. 161) hetween the points of suspension, and the depth DCof t^
lowermost point, he given, hang one extremity of a fine imifa''^
chain or cord at one of the points A, and (letting the chain or e^
adjust itself as a festoon in a vertical plane) lengthen or shorten^
as it is held near the other end, over a pin at B, until^ when at r^
CHAP. VII.] CUBVBfi: CATBHAHr. 183
it just reacbes the point C : so shall the cord form the catenary ; and
a pencil passed along the cord, from A by a, C, &, to B, will mark the
curve upon a vertical board brought into contact with it.
13. All catenaries that make equal angles with their ordinates at
their points of suspension are similar, and have j; to y a constant
ratio : and of any two which do not make equal angles, but have
X to y in different ratios, a portion may be cut from one curve similar
to the other. Thus, let ACB and A'C'B' (fig. 162), be the two
curves, of which A' C" B' is the flatter. Suppose them placed upon
one axis D C C\ and the tangent T' S', to the lower curve, at B', the
point of suspension, to be drawn. Then, parallel to T'S' draw
another line TS to touch the other curve in b. Through b draw
ba parallel to B^A^. So shall the portion aCb of the upper catenary
be similar to the lower catenary A'C'B\
14. With reference to the practical uses of the catenary, we may
now blend the geometrical and the mechanical consideration of its
properties. Taking any portion Cb (fig. 163) of the catenary, from
the lowest point C; its weight may be regarded as supported by
tensions acting in the tangential directions C N, by. The strains at
C and b may be conceived as acting at the point of intersection N ;
above which, therefore, in the vertical direction N O, the weight of
the portion Cb may be conceived to act at its centre of gravity, G.
Hence, strain at C : weight of C^ :: sin ON6 : sin &NC :: cos&NR
: sin 6 N R :: radius : tan &NR :: radius : tan </^ N.
Hence, the horizontal tension at C being constant^ the weight, and
consequently the length of any portion ci of the uniform chain must
be proportional to the tangent of the inclination of the catenary to
the horizon at the extremity b of the said portion. This may he
ttgarded as the characteristic property of the catenary,
15. In like manner, the horizontal strain at C : oblique strain at
& :: sin ^ N O : sin C N O :: cos & N R : radius :: radius : sec & N R.
Therefore, the strain exerted tangentially, at any point 6, is pro
portional to the secant of the inclination at that point.
Also, from § 14 and § 15, tangential strain at b : weight of B & C ::
wecdby : tani^N.
These properties evidently accord with the preceding equations.
16. Let, then, CO, in the axis produced downwards, be eqnal to
the parameter, or the measure of tne horizontal strain at C ; and upon
O as a centre with tadius CO dcscHbe a circle. A tangent dt drawn
to this circle from cf, will be parallel to the tangent &NA of the
curve at the point b to which db ib the ordinate. That tangent dt
(to the circle} will also be equal in lencth to the corresponding por
tion 5 C of die curve : while the tension at b will be expressed by
a length of the chain equal to the secant O d. So again, if D T be
a tangent to the circle drawn from D, it will be equal in Icneth to
B5c, and parallel to the tangent to the catenary at B; while the
secant OD will measure the oblique tension at B; evidently, ex
eeeding the constant horizontal tension or strain at C, by the
abadna CB.
184 CUBVBS: CATENAHY. [pART I.
17. When the parameter of the catenary, or the line which mea
sures the tension at the lowest point, is equal to the deflexion DC;
if each of these he supposed equal to 1, then A B = 12*6339, the
length of chain A C B = 3*4641 ; the strain at the points of suspen
sion A and B will each he 2, that at the lowest point heing 1 ; and
the chain at A and B will make an angle of 60^ with the horizon.
18. If the strain at C he equal to the weight of the chain, and each
denoted by 1: then AB = *96242, DC = 1180340, the tension
at A or B = 1*118, the angle of suspension at those points nearly
26° 34'; the width of the curve is 8*1536 times, and the length
8*4719 times, the deflexion DC.
19. If the strain or tension at the lowest point be doable the
weight of the chain : then if the parameter be 1, A B will be '49493,
C D = 03078, the strain at A or B 1*03078, the angle of suspenaon
about 14° 2', the width or span 16*0816 times, and the length of
chain 16*2462 times the deflexion.
The magnitudes of the lines, angles, and strains in many other
cases, may be seen in the table below. The whole theory may be
yerifled experimentally, by means of spring steelyards applied to a
chain of given length and weight, placed in various positions, accord^
ing to the method suggested at page 244, when trc»Etting of the me
chanical powers.
20. Taking A B = </, C D = A, length A C B = /!» strain at C or
parameter = p, then, in all cases where the deflexion is small com
pared with the length of the chain. Professor Leslie shows*, that
d' d
p =  , f ] A .... strain atAorB = rflA
or /? = — — I ^ .... strain atAorB=7 + JA
^ = ^■^33
In this case, too, the strains at G and A or B are nearly in the
inverse ratio of the deflexion t.
21. The following table is abridged from a very extensive one
given by Mr. WarCy in his " Tracts on Vaults and Bridges." Two
examples will serve to illustrate its use.
Ex. 1. Suppose that the span of a proposed suspension bridge is
to be 560 feet, and the deflexion in the middle 25^ feet; what will
be the length of the chain, the angle of suspension at the extemities,
and the ratio of the horizontal pressure at the lowest point, and the
oblique pressures at the points of suspension, with the entire weight
of the chain ?
* Elements of Natural Philosophy, p. 63.
f For a very complete investigation uf the proper forms of catenaries for tns
pension bridges, with remarks on the Menai Bridge, and on the failure of the
suspension bridge at Broughton, see Mr. Eaton Hodgkinaon*s paper in the
Memoirs o( the Manchester Society, vol. v., New Series.
CHAP.
VII.]
CURVXS: CATBNARY.
18^
TaUe ofBdaiions of Catenarian Curves^ the Parameter being de^
notedly 1.
wiSS^
Dtfloioii.
or DC.
8«»^.or
Laofth ot
chain, or CbD.
TAMkmat
the point at
Semiipan di.
Tided "by the
deflexion, or
DB fiC.
!• C
•00015
•01745
01745
10001
114586
2
•00061
•08491
03492
10006
67279
8
•00187
•05288
•05241
10014
88171
4
•00244
•06987
•06993
10024
28613
5
•00882
•08788
•08749
10088
22874
6
•00551
•10491
•10510
1'0055
19046
7
•00751
•12248
•12278
1'0075
16309
8
•00988
•14008
•14054
1*0098
14'254
9
•01247
•15778
15838
10125
12654
10
'01548
•17542
17683
1'0154
11872
11
•01872
19818
•19438
10187
10820
12
•02284
21099
•21256
10223
9444
18
•02680
•22887
•28087
10263
8701
14
03061
•24681
24983
10806
8062
15
•08528
•26484
•26795
10853
7608
16
•04080
•28296
28675
10408
7021
17
04569
•80116
•80573
• 10457
6691
18
•05146
•81946
32492
10515
6208
19
•05762
•88786
34433
1*0576
5868
20
•06418
•85687
36897
1'0642
5563
21
•07114
•87502
•38386
10711
5271
22
•07858
•89876
40408
10786
5014
28
•08686
•41267
42447
10864
4778
24
•09484
•48169
•44528
1^0946
4562
26
•10888
•45087
•46681
11084
4361
28
•11260
•47021
•48778
1'1126
4176
28
•18257
•50940
•58171
11826
8843
80
•15470
•54980
•57785
ia547
8561
82 4
•18004
59120
•62649
11800
8284
84 16
•21003
•68710
•68180
1'2100
8034
86 52
•24995
•69820
•74991
12499
2773
89 11
•29011
•74480
•81510
12901
2567
41 44
•84004
•80290
89201
18400
2362
44
•89016
•85660
•96569
1'3902
2196
46 1
•48999
•90660
1*03610
14400
2060
48 11
•49981
•96230
111780
14998
1925
50 8
•56005
101420
119740
15800
1811
52 9
•62978
107060
128690
16297
1699
54 18
•71021
118040
188740
17102
1592
56 28
•81021
119950
150890
18102
1481
58 8
•88972
125100
160340
18897
1416
60
1^00000
181690
178210
20000
1817
64 6
128940
147020
205940
22894
1140
67 28
160950
161350
241020
26095
1002
67 82
161680
161640
1
241820
26168
09998
186 curves: catenary. [part i.
Here D B — D C = 280 h "25 875 = 10'8*2, a namber which is
to be foand in the table.
Opposite to that namber^ we find 11° for the angle of saspension,
DB= 19:318, CB= 19488, tension at A or B = 10187, the
constant tension at the Tertex being 1. (Fig. 161).
CoDseqnentlj, 19318 : 19438 :: 560 : 563*48 length of the chain.
Also, horizon, pressure at C being taken as 1*0000
the oblique pressure at A or B will = 1*0187
and the entire weight of chain will = '39876
Ex. 2. Sappoae that while the span remains 560, the deflexion is
increased to 51.
Here D B f D C = "284) ; 51 = 549. This number is not to he
found exactly in the table. The nearest is 5*553 in the last column,
agreeing with *20^, the angle of suspension.
Now, 5*55 — 5*49 = 06, and 5*55 — 5'27 = '28, the former
differenee being oearlj one«>fifUi of the latter. Hence, adding to
each number, in the line agreeing \iith '20°, onefifth of the differ
ence between that and the corresponding number in the next line,
we shall have
An^e of suspension = aO"" 1*2', DC = 06556, DB = 86010,
CD = 36797, tension at A = 1*10656.
Hence 36010 : 36797 :: 560 : 572 24, length of chain.
Also, horixontal pressure at C being taken as 1*0000
the oblique prest^ure at A or B will ss 1*10656
and the entire weight of chain will = 73594
Comparing this Tilth the former case, it nill be seen that the
tensions at C and A, in reference to the weight of the chain, arc
diminished nearly in the inrerse ratio of the two values of DC;
thus confirming the remark in art. '20.
In making use of this table, the remark at page 1 80 must be borne
in mind, since the results obtained from it, will be only approxima
tions more or less correct, as the weight of the chain exceeds that of
the roadway and load. Where greater accuracy is required, the
formula given at page 208 must be employed in its stead.
PT. II. CH. I.] mechanics: 8TATIC8. 18?
PART II.
MIXED MATHEMATICS.
CHAP. I.
MIOHANIOS IN OBNBRAL.
1. Meehanies is ^e science of equilibrium and of motion.
2. Every cause which tends to move a body, or to stop it when in
motion, or to change the direction of its motion, is callea a force or
3. The dkreetion of a foree is that straight line in which the point
to which the force is applied tends to move by virtue of that force.
4. When the forces that act upon a body, destroy or annihilate
each other s operation, so that the body remains quiescent, they are
«id to be in tquiUhrium^ and are then called presntres,
5. The dirmticn of a pressure is the straight line in which that
presaore tends to prevent the motion of the point to which it is ap
pKad.
6. The effect of anv force or pressure is found to be the same, at
wfaattfver point in the line of its wection it be applied.
7. Two forces or pressures are said to be tqualy when, being ap
plied in appoiite directions to the same point, no motion ensues. If,
however, both forces are supposed to act in the iame direction, the
single force or pressure which would be required in the opposite
direction to keep the point in equilibrium, is said to be double either
of the former forces. And one of the former forces being taken as
the unit, the latter force would be represented by two of such units.
8. When two forces can be expressed in terms of the same
unit, the^ are said to be eommensurcAle.
9. It IS usual to represent forces and pressures by lines, the diree
iioH of the line coinciding with the actual direction of the force, and
(a line of a certain length being taken as the representative of a unit
of force,) the ien^ of the line expressing the amount or magnitude
of the given force or pressure ; and a force thus denoted is said to be
represented both in magnitude and direction,
10. When several forces or pressures act in different directions
upon the same point, it is possible to find the direction and magnitude
of another force or pressure which would replace all the others ; that
188 mechanics: statics. [pabt ii.
is, if the others were remored, and this one force applied in their
stead, the effect produced would he the same. Sach a force is called
the resultant of the others, which are named the components^ and the
operation which we have supposed is called the composition of forces,
and its converse, or finding the directions and magnitudes of any
nnmher of forces which would produce the same effect as any one
given force, is called the resolution of forces.
11. The moment of a force or pressure ahont any point, is the
perpendicular distance from that point to the line of direction in
which the force acts, multiplied hy the numher of units expressing
that force.
12. Vis inertia^ or power of inactivity, is defined hy Newton to
be a power implanted in all matter, by which it resists any change
attempted to be made in its state, that is, hy which it requires force
to alter its state, either of rest or motion.
13. Vis vioiiy or living force, a term used by Leibnitz to denote
the force or power of a body in motion; or the force which would
be required to bring it to a state of rest.
14. Mechanics is usually divided into five branches; viz. —
I. Statics^ which relates to the equilibrium of pressures applied
to solid bodies, and of the weight and pressure of bodies when
at rest.
II. Dynamics^ which relates to tlie motion prodaced in solid
bodies by the application of force.
III. ffydrostaticsy which relates to the equilibrium and pressure
of nonelastic fluids, and of the weight, pressure, and sta
bili^ of bodies immersed in them.
IV. NydrodynamicSy which relates to the motion of non'elaaie
fluids by the application of force.
V. Pneumatics^ which relates to the various circumstances at
tending the equilibrium or motion of elastic fluids.
CHAP. II.] STATICS. 189
CHAP. II.
STATICS.
Sect. I. Statical Equilibrium,
1. Ip any two pressures applied to a point keep it in eqailibrinro,
they most be eqnal to each other, and must act in the same straight
line and in contrary directions.
3. If any tkree pressures applied to a point keep it in equilibrium,
they must all act in the same plane ; and any one of those pressures
is represented in magnitude and direction by the diagonal of a paral
Ukaram^ whose sides represent the other two pressures in magnitude
and direction.
3. If nnj/our pressures whose directions are not in the same plane,
applied to a point, keep it in equilibrium, any one of those pressures
is represented in magnitude and direction by the diagonal of a paral^
lelapipedon^ whose contiguous edges represent the other three pres
sures in magnitude and direction.
4. If any number of pressures whose directions are all in the
same plane, applied to a point, keep it in equilibrium, those pressures
will be represented in magnitude and direction by a polygon whose
M&i are made parallel to the direction of those pressures, and pro
portional to their magnitude.
5. If any number of pressures in the same plane be in equili
brium, and any point be taken in that plane from which their mo
ments are measured, then the sum of the moments of those pressures
which tend to turn the plane in one direction about that point, will
be equal to the sum of the moments of those which tend to turn it
in the opposite direction.
6. If any number of pressures acting in the same plane, and which
are in equilibrium about a given point, be moved parallel to the direc
tions in which they act, until they all intersect in any given point,
they will still be in equilibrium about that point.
7. The resultant of two parallel pressures, if acting in the same
direction, is equal to their 8um; but, if acting in contrary directions,
is equal to their difference^ the direction of their resultant being in
the direction of the ereater pressure. And generally the resultant of
any number of parallel pressures is equal to the several pressures
added together with their proper signs.
190 STATICS. [part II.
From the foregoing propositions we may deduce the following for
mulffi: —
8. Let P. and P^ represent as j two pressures, and ff the angle
formed hy tneir two lines of direction ; kt R be their resultant ; y
the angle which its line of direction makes with that of Pj, and ^ that
which it makes with P^; then
R = >/Pi* + Pj' + 2 Pj Pj cos /9 . . . (I.)
P.. sin $ , .
tan y =  — ~ ^ (II.)
'^ Pi + P« cosg ^ ^
Pj : Pg :: sin ^ : sin y (IH.)
or, three forces being in equilibrium, any two of them are to each
other inversely as the sines of their inclinations to the third.
9. To determine the resultant of any number of pressures in the
same plane: — Let P,, P^, P,, &c. represent the pressures in magni
tude, and a J, a^, a 3, the inclinations which their lines of direction
make with some line given in position ; let R be their resultant, and
» its inclination to the same line; then
R cos » = P cos a, f Pg cos aj f .... + P« COS «« . (IV.)
R sin » = Pj sin a, + P^ sin a^ + . . . . + P« sin «, . . (V.)
in which the several terms are taken positive or negative according
to the direction in which they act. From these we obtain
R = {>/(P, sinai+Pjsina, + + P,sin«,)* +
(Pi cos a, + Pg cos «^ f + P, cos aj«} . . . (VL)
t^„,^PBina, 4P.8ina,.f.... + P,«i°a, ...(VU.)
P, cos a J 4 Pg COS«j 4 . . . . h P^ COS «,
10. Let Pj, Pg, P., &c., be any number of parallel pressures,
(being + or — according to their direction,) and D,, D^, D^, Ac,
be their perpendicular distance from any given parallel plane : let B
be their resultant, and A its perpendicular distance from the same
plane: then
R = P, + P2 f + P (VIII.)
^^P»I>>iP,D,+ 4P,D,
Pi + P« + . . . h P. ^ '
As these formulae and propositions are of universal application in
the constructions of civil engineers, architects, and mechaniciaos, we
shall give a few simple examples, to render tlieir use clear to those
who are not familiar with mathematical form'ulee.
Ex, 1. Suppose that a weight B is attached by a stirrup to the foot .
of a kingpost A B, which is attached to two rafters A C, A D, in the
respective positions shown in fig. 1 64. Then if A E be set off upon
CHAP. II.] STATICS. 191
AB, equal in numerical value to the yertical strain upon A B, and the
parallelogram AFEO be completed, AF measured upon the same
scale will show the strain upon the rafter A C, and A O the strain
upon A D.
Ex. 2. Let it be proposed to compare the strains upon the tie
beams AD, and the struts AC, when they sustain equal weights B, in
the two different positions indicated by figs. 165 and 166. Let A £ in
one figure be equal to the corresponding yertical line A£ in the
other, and in each represent the numerical value of the weight B,
that hangs from A. Through E in both figures, draw lines parallel
to D A, A C, respectively, and let them meet A C, and D A produced
in F and O: then AFEG in each figure is the parallelogram qf/orces
by which the several strains are to be measured. A G represents the
tension upon the tiebeam A D, and A F the strain upon the strut
A C. Both these lines are evidently shorter in the lower figure than
they are in the upper, A E being of the same length in both : there
fore the first figure exhibits the least advantageous position of the
beams. It is evident also, that while C A tends upwards and D A
downwards, the greater the angle D A C, the less is its supplement
C A G, and the less the sides F A, A G, of the parallelogram.
JSm. 3. Let it be required to determine geometrically, and by com
potadon, the magnitude and direction of the four pressures P^, P^,
Pj and P^, (fie. 167,) all applied to the point A, and acting in the
sme plane; let Pj = 12, P^ = 9, P, = 16, P^ = 15, and the
angles which their directions make with the line BAG, (given in
pontion,) be 77^ 37% 9° and 48% respectively.
1. CrecmeiricaUy. — From the point A (fig. 168) lay off the angle
CAD equal 77% and upon the line A D set off 12, from any scale of
equal parts ; then through D draw the line D E, equal to 9 parts by
the same scale, and making an angle of 37° with B A C ; again,
through £ draw the line £ F equal to 16, and inclined 9° to B A C ;
and lastly, through F draw £ G, equal to 15, and inclined 48° to B A C;
then the straight line AG, connecting the first and last points, and
completing the polygon A D £ F G A, will be the resultant of the four
given pressures ; and being measured by the same scale, will be found
equal to 36*72 parts nearly, and to be inclined 13° to B A C.
2. By Computation.— k^ the pressures P^, Pg) and P., are not on
the same side of the line BAC (fig. 167) as P^, the signs of their
ftfies most be taken differently ; but as they are ail on the same side
of the line H A I, (perpendicular to B A C,) the signs of their cosines
must be taken alike; then if R is their resultant, and » its inclination
to B A C, we obtain from formula (VI.) —
E=^/{(ia X 9744 + 9 X 6018+ 16 X 166415 x •7431)' f
(12 X 225 + 9 X 7986 f 16 x 9877 + 16 x 6691)*} = 367138;
•ad from formula (VII.) —
12 X 9744 + 9 X 6018 + 16 x 1564  15 x '7431 ^^,,,^
ttn«B3 . ='23106
12 X 225 + 9 X 7986 + 16 x 9877 + 13 x 6691
.. • = 13^
In which the values are the same as were found geometrically.
192 CENTRE OF GRAVITY. [PABT II.
Sect. II. Centre of Gravity,
1. Gravity is the force in virtue of which bodies left to themselves
fall to the earth in directions perpendicular to its surface.
2. We may distinguish between the effect of gravity and that of
weighty by observing that the former is the power of transmitting to
every particle of matter a certain velocity which is absolutely inde
pendent of the number of material particles ; while the latter is the
effort which must be exercised to prevent a given mass from obeying
the law of gravity. Weighty therefore, depends upon the fiuu$; but
gravity has no dependence at all upon it.
3. The centre of gravity of any body, or system of bodies, is that
point about which the body or system, acted upon only by the force
of gravity, will balance itself in all positions: or it is a point which,
when supported, the body or system will be supported, however it
may be situated in other respects.
The centre of gravity of a body is not always within the body itself;
thus the centre of gravity of a ring is not in the substance of the
ring, but in the axis of its circumscribing cylinder; and the centre of
gravity of a hollow staff, or of a bone, is not in the matter of which
it is constituted, but somewhere in its imaginary axis; every body,
however, has a centre of gravity, and so has every system of bodies.
4. Varying the position of the body will not cause any change in
the relative position of the centre of gravity ; since any such muta
tion will be nothing more than changing the directions of the forces,
without their ceasing to be parallel ; and although the amount of the
forces may not continue the same, in consequence of the body bein?
supposed at different distances from the earth, still the forces upon aO
the molecules vary proportionally, and the position of their centre re
mains unchanged.
5. When a heavy body is suspended by any other point than its
centre of gravity, it will not rest unless that centre is in the same
vertical line with the point of suspension ; for in all other positions
the force which is intended to ensure the equilibrium will not be di
rectly opposite to the resultant of the parallel forces of gravity upon
the several particles of the body, and of course the equilibrium will
not be obtained. (See Art. 9, on Pendulums^ page 222.)
6. If a heavy body be sustained by two or more forces, their
directions must meet, either at the centre of gravity of that body, or
in the vertical line which passes through it.
7. When a body stands upon a plane, if a vertical line passing
through the centre of gravity fall within the base on which the body
stands, it will not fall over; but if that vertical line passes without the
ba^e, the body will fall, unless it be prevented by external means.
When the vertical line falls upon the extremity of the base, the body
may stand, but its state (which is called unstable equilihriuni) may
be disturbed by a very trifling force ; and the nearer this line passes
to any edge of the base the more ensily may the body be thrown
CHAP. II.] CBNTRB OF GRAVITY. 193
OTer ; the nearer it falls to the middle of the hase, the more firmly
the body stands.
Upon this principle it is that leaning lowers have been bnilt at Pisa,
and various other places ; the Tertical line of direction from thecentre
of gravity falling within the
base. And, from the same
principle it may be seen , ^^^
that a waggon loaded with ^fl^p^^
heavy materials, as B, may
stand with perfect safety, on
the side of a convex road,
the vertical line from the
centre of gravity falling be
tween the wheels ; while a waggon A with a high load, as of hay, or
of woolpacks, shall fall over, because the vertical line of direction
fells without the wheels.
8. Owing to the great distance of the earth's centre from its sar
&ce, the directions of the force of gravity of the several molecule
composing a body may be considered parallel without any appreciable
error, and therefore all that has been said in the foregoing section on
the subject of parallel pressures will apply to the force of gravity ;
and the formula for finding the position of the resultant of any num
ber of pandlel pressures, may be applied to determine the position of
the centre of gravity.
Thas, if Bj, B^, B3, &c., denote the weights of the particles of
any body, and D^, D„, D,, &c., the perpendicular distances of their
reflective centres of gravity from any given plane ; then, the dis
tance TA) of their common centre of gravity from the same plane is
found by formulsB (IX.) to be
^ ^ B^D, +B^ Dg I +B,D,
Bj +B2 + +B.
Therefore, if by means of this formula we ascertain the distance of
the centre of gravity from any three planes given in position, we shall
have determined its exact situation in space.
9. The common centre of gravity of two bodies divides the right
line which joins their respective centres of gravity, in the inverse
ratio of their weights. The centre of gravity of any number of bodies
may be found, by finding the common centre of gravity of any two of
the bodies, and then considering this as the centre of gravity of a
body equal to the combined weight of the two, we may find the com
mon centre of gravity of this imaginary body and a third ; and thus
proceed, ad libitum,
10. If the particles or bodies of any system be moving uniformly
and rectilineally, with any velocities and directions whatever, the
centre of gravity is either at rest, or moves uniformly in a right line.
Hence, if a rotatory motion be given to a body and it be then left
to move freely, the axis of rotation will pass through the centre of
194 CIKTRB OF ORAYITT. [PABT H.
grwnty : for that centre, either remaining at rest or moving nniformly
forward in a right line, has no rotation.
Here too it may be remarked, that a forte appUed at the centre of
^ravit^ of a body^ eannct produce a rotaUny motion.
11. To find the centre of gravity meebanically, it is only requisite
to dispose the body successiYely, in two positions of eqoilibriam, by
the aid of two forces in vertical directions, applied in succession to
two differMt points of the body; the point of intersectkm of these
two directions will show the centre.
This may be exemplified by particularising a few methods. If the
body have plane sides, as a piece of board, hang it up by any point
(A fig. 169), then a piuiAbline suspended from the same peine will
pass through the centre of gravity; therefore toark that line (AB)
upon it : and after suspending the body by another point (C), apply the
plummet to find another such line (C D) ; then will th^ interaeetion
show the centre of gravity.
Or thus : hang the body from a tack successively by two strings
attached to different parts of it, and each time mark upon it the^line of
a plummet attached to the same tack ; then will the intersection of the
two lines be the centre of gravity.
Another method : Lay the body on the edge of a triangular prism,
or such like, moving it to and fro till the parts on both sides are in
equilibrio, and mark a line upon it close by the edge of the prism:
balance it again in another position, and mark the fresh line by the
edge of the prism; the vertical line passing through the mterseetion of
these lines, vrill likewise pass through ^e centre of gravity. The
same thing may be effected by laying the body on a table, till it ie
just ready to fall off, and then marking a hue upon it by the edge of
the table : this done in two positions of the boay will in like manner
point out the position of the centre of gravity.
Wlien it is proposed to find the centre of gravity of the arch of a
bridge, or any other structure, let it be laid down accurately to aoale
upon pasteboard ; and the figure being carefully cut out, its centre of
gravity may be ascertained by the preceding process.
12. The centre of gravity of a right line, or of a paraUdlogram,
prism, or cylinder, is in its middle point; as is also that of a circle, or
of its circumference, or of a sphere^ or of a regokr polygon ; the
centre of gravity of an ellipse, a paraboks a cone, a conoid, a sj^
roid, &c., is situated in its axis. And the same of all symmetrioal
figures.
13. The centre of gmvity of a triangle is the point of intersectioa
of lines drawn from the three aneles to the middles of the sides re*
spectively opposite: it divides ea(m of those lines into two portions ia
tlie ratio of 2 to 1.
14. In a Trapemum. Divide the figure into two triangles by the
diagonal AC (fig. 170) and find the centres of grarity £ and F of
these triangles; join £ F, and find the common centre O of these twe
by this proportion, ABC : ADC :: FO : EG, or ABCB : ABC ::
£F : £0. Or, divide the figure into two triangles by a dii^nal
CHAP. II.] CmRB OP GRAVITY. 195
BD; then find thm centres of gravity; and the line which joins
them will intersect £F in O, the centre of grayity of the trapezium.
15. In like manner, for any other plane fignre, whatever he the
numher of sides, divide it into several triangles, and find the centre
of gravity of each ; then connect two centres together, and find their
common centre as ahove; then connect this and the centre of a third,
and find the common centre of these; and so on, always connecting
the last found common centre to another centre, till the whole are
included in this process; so shall the last common centre be that
which is required.
16. The centre of gravity of a circular arc is distant from the
centre of the arc a fourth proportional to the arc, the radius, and the
chord of the arc.
17. In a circular sector, the distance of the centre of gravity from
2 c r
the centre of the circle is— — ; where a denotes the arc, e its chord,
oa
and r the radius.
18. The eentres of gravity of the mrfaees of a cylinder, of a cone,
and of a conic frustum, are respectively at the same distances from
the origin as are the centres of mvity, of the parallelogram, triangle,
and trapezoid, which are verticsl sections of the respective solids.
19. In the segment of a sphere, or spheroid, A being the whole
ans, and k the height of the segment, the distance of the centre of
A A ._ dk h
gravity from the vertex is equal ^^ 77 ; and when the height be
o A — 4 A
oomes half A, or the segment becomes a hemisphore, the distance
Irom the vertex is equal to ^ A.
20. The centre of grarit^ of the surface of a spheric segment is
at the middle of its versed sine or height.
21. The centre of gravity of the convex turface of a spherical
lone, ia in the middle of that portion of the axis of the sphere which
is intercepted by the two bases of the zone.
22. In a cone, as well as any other p3rramid, the distance from the
vertex is \ of the axis.
23. In a conic frnstnm, the distance on the axis from the centre
of the leaser end, is J A. j : whore h equals tlie
he^fat, and R r the radii of the greater and lesser ends.
24. The same theorem will serve for the frustum of any regular
pyramid, talcing R and r for the sides of the two ends.
25. In the paraboloid, the distance from the vertex is f of the
axis.
26. In the frustum of the paraboloid, the distance on the axis from
2 R* + r*
the centre of the lesser end, is J h.  : where h equals the
hcigfat, and R r the radii of the greater and lesser ends.
2
196 IQUILIBBIUM OF PIBB8. [PABT H.
27. Every figure, whether superficial or solid, generated bj the
motion of a line or surface, is equal to the product of the generadng
magnitude multiplied bj the path of its centre of gravity.
As an example, let A B C (fig. 171) be a right angled triangle, the
revolution of which on the leg A B will produce a cone ; let A B =: 9,
BC = 6, and D be the centre of mvity; then by § 13, AD ^  AF,
therefore D£ sfFB = Q equal the radius of the circular path de
scribed by the centre of gravity, the circumference of which will
therefore be 4 x 31416 = 125664, which multiplied by (3 x 9)
= 27, the area of the triangle, will be 339*2928, the solid content of
the cone; and by the rule given at page 130 for finding the solidity
of a cone, we obtain 12* x 7854 x 3 = 3392928, the same
result.
Sect. III. General application of the principles of StaUcs to
the equilibrium of Structures,
Every structure is exposed to the operation of a system of forces;
so that the examination of its stability involves the application of the
general conditions of equilibrium.
Now, no part of a structure can be dislocated, except it be either
by a progressive, or a rotatory motion. For either the part is dis
placed, without changing its form^ in which case it is (as a system of
invariable form) incapable of receiving any instantaneous motion,
which is not either progressive or rotatory ; or else it happens to be
displaced, changing at Vie same time its form^ which, considering the
cohesion of its parts, cannot take place without its breaking in its
weakest section ; in which case a progressive motion is generated if
the force acts perpendicularly to the section, and a rotatory motion,
if it acts obliquely.
We shall here consider the most useful cases ; indicating by the
word stress^ that force which tends to give motion to the structure,
and by resistance^ that which tends to hinder it.
EQUILIBRIUM OF PIERS, OR ABUTMENTS.
1. If we suppose figure 1 72 to be the vertical section of a pier, we
may reason upon that section instead of the pier itself, if it be of
uniform structure.
Let O be the place of the centre of gravity of the section A B C D,
S Z the direction in which the stress acts, meeting X I, the verticai
line through the centre of gravity, in I. Then, supposing the stress
to be resolved into two forces, one (Pj) vertical, the other (P^)
horizontal ; the pier (regarding it as one bodv) can only give way
either by a progressive motion from B towards A, or by a rotatory
motion about A.
2. The progressive motion is resisted by friction. If W denote
the weight of the pier, Pj the stress estimated vertically, and Pj its
horizontal effort, then the pressure on the base sW + P^, and
CHAP. II.] EQUILIBRIUM OF PIERS. 197
potting /for the coefficient of friction, its friction =/(W + Pj,
which 18 the amount of the resistance to progressive motion. So that
to ensure stahilitj in this respect we must have
/(W + PJ> P, (I.)
8. To ensure stahilitj in regard to rotation, taking the moments of
the forces ahout the point A, we must have
W . AX f Pj . Xe > P, . Tq (II.)
Or, supposbg the stress not to he resolved, its moment ahout tho
point A must be less than that of the weieht of the pier about the
same point ; that is, putting S for the whole stress,
W.AX>S.aY (III.)
Or, by a graphical process, suppose the two forces W and 8 to he
applied at I, and complete the parallelogram, having sides which
represent these forces. Then must the diagonal representing the
resultant of those forces produced meet the base on the side of A,
towards B, to ensure stability.
4. If, as is very frequently the case, the vertical section of the
pier is a rectangle, putting k for the height of the pier, h for its base,
w for the weight of a cube unit of the material of which it is formed,
and R a horizontal stress applied at the summit of the pier ; then
the pier will be in a state of unstable equilibrium, as far as regards
its progressive motion, when
hfhw^lBi :.... (IV.)
and as r^grds its being overthrown when
V=* (V)
But in order that the structure may really be secure, these equations
most become
b/hw>B^ and — >R,
and the more the first member exceeds the second the greater will
be the stability of the pier.
EwampU 1. Suppose a rectangular wall 89*4 feet high, and of a
material weighing 125 lbs. per cube foot, is to sustain a horizontal
atnin of 99(K) lbs. avoirdupois on each foot in length, applied at its
somiDit: what must be its thickness that there may be an equilibrium,
taking /= 1*75.
By transposing formula (IV.) above, we obtain
and substituting the several values given above, we have
h = = 268 feet;
•75 X 894 X 125
198 PRESSURI OF EARTH AOAlVWt WALLS. [PABT II.
therefore the thickness of the pier at its base mast be more than
2*68 feet, otherwise it will be moved horizontally by the stress R.
But we must also inquire, what thickness is requisite to prevent the
pier being overthrown; and this we shall ascertain from formula
(v.), which by transposition becomes
V2R
/2 X9900 __^
and * = A / — r^T — = 12*58 feet.
Therefore, we see, that the pier most be more than 12*58 feet
upon its base, otherwise it will be overthrown by the horizontal
strain at R.
Here, as the thickness required to prevent overturning is much
the greatest, the computation m reference to the other kind of equili
brium may usually be avoided.
Example 2. An embankment, or dam, A B D (fig. 17d), formed
of clay between two rubble walls, is exposed to the pressure of
80 feet of water ; it is required to ascertain whether the dam will be
overthrown, and also whether it will slip upon its bed. Its dimen
sions are, D C = 4 feet, B C which is vertical = 32 feet, and A B
Bs 12 feet; its weight may be taken at 135 lbs. per cubic foot, and
the coefficient of friction or f ^ '5.
First, by the method laid down in § 14, page 194, we ascertain
the distance of the centre of gravity of the wall from its interior face
B C, to be 4*333 ; therefore the distance (A X) of its line of direc
tion from A will be 7*667 feet. The weight of the wall (W) will be
12+4
—  — X 32 X 135 = 34560 lbs.; the pressure of the water
30
(Pg or 8) will be — X 62*5 = 937*5, and the centre of pressure
of the water being at twothirds of its depth (as explained at page 250),
the distance (AZ) of its line of direction above the point A will
be 10 feet.
Now, in order to ascertain its stability to resist progressive motion,
we must substitute these values in formula (I.), when we obtain
•5 X 34560 > 937*5, or
17280 > 9375;
therefore there would be no fear of the embankment slipping upon
its base. And from formula (III.) we have,
34560 X 7667 > 9375 X 10, or
264972 > 9375 ;
therefore the wall cannot be overthrown.
PRE88URB OF EARTH AGAINST WALLS.
1. Let DACB ^fig. 174) be the vertical section of a wall behind
which is posited a oimk or terrace of earth, of which a prism whose
CHAP, n.] raammR of xabth against walls. 199
section is represented by CBH wonid detach itself and fall down,
were it not preyented by the wall. Then B H is denominated the
line of rupture or the natural slope^ or natural deeltvity. In saudy
or loose earth, the angle CHB seldom exceeds 30°; in stronger
earth it becomes 37°; and in some favourable cases more than 45°.
2. Now, the prism whose vertioal section is C B H, has a tendency
to descend along the inclined plane B H by reason of the force of
gravity ; bat it is retained in its place, not only by the force opposed
to it fy the wall, bat also by its cohesive attachment to the face B H,
aDd by its fiicdon upon the same surface.
If we resolve the weight of this prism into two forces, one acting
pcfpendicular to the plane B H, and the other parallel to the same,
the latter minus the force required to overcome either the cohesion
or the friction of the surface B H, will be the strain acting in a direc
tion passing through the centre of gravity of the prism CBH and
parallel to B H, upon the back of the wall, and tending to overthrow
It ; and the aiBount of this atraio, and its direction, being ascertained,
the formulsB already given may be employed to determine the sta
InU^of the wall,
I H 2s LH + eohesioii + friction.
S. It is evident, therefore, that m the angle at whidi the earth
will stand U one <>f the elements in the calculation, the solution to
this inquiry must be, in a great measure, experimental. It has been
found, however, theoretically, bv M. Prony*, and confirmed experi
mentiJly, that the angle formed with the vertical (CBH) by the
prism ojf earth that exerts the greatest horizontal stress against a
wall, is half the angle which the natural slope of the earth makes
with the vertical : and this emrious result greatly amplifies the whole
inquiry.
Puttisg A for the heiffht of the wall, ff for the angle CBF, or
half C B H, the natural slope of the ground, w^ for the weight of a
cabit onit of the ground, and /for the coefficient of friction we have
h^w^ . tan/S_
2
the weight of the prism CBF; and resolving this into two forces,
one (P,) perpendicular, the other (P^) parallel to BH, we have
for the former
and for the latter
J^w^ .imp .m$ ^
5 P,,
2 "" ^
* See a denxmstnetion at p. 309, vol. il. tenth edition of Dr. Hatton't
ConiM of MatbcmiitiGt.
200 PRBS8URB OP EARTH AGAINST WALLS. [PART 1% '
Therefore tlie friction along the surface B H, will be
fh^w^ . ton /9. sing y, .
which subtracted from P^, gives
(l_/.tang)*!i^l^=S (VII.)
equal the strain acting through the centre of gravity of the prism^i
upon the back of the wall ; now, as the centre of gravity of a triangle^
is situate at a third of its height, and the direction of the strain S i^
parallel to the sloping base B H, it will meet the wall at onethird or"
its height, and b being put for the breadth of the base of the wall,
we shall have
st^.sinpb.coBp\ (VIII.)
equal the moment of the strain S about the point A, tending to
overthrow the wall.
Example, Suppose a wall is to be built of brickwork (weighing
117 lbs. per cubic foot), to support a terrace 39 feet in height, the
earth composing which weighs 105 lbs. per cubic foot, its natural
slope being 53° from the vertical, and the coefficient of friction *45 ;
it is required to find the breadth which must be given to the wall at
its base in order to ensure its not being overthrown.
By substituting these values in formula (VIIL), we obtain
S = (1  45 X 4986) «^' X iO^ X '^^^^ = „630;
the moment of which from formula ( VIII.) is
27636 f^ X 4462  ^ x 8949^ = 160322 — 24731 . b.
Therefore by formula (III.), we have
^^^^ "^ > 160322  24731 b, or
b > 458 ;
that is, the base of the wall must be more than 4*58 feet, in order
that it shall not be overthrown by the pressure of the earth.
4. Of the experimentol results, the best which we have seen are
those of M. Mayniel, from which the following are selected; in all of
which the upper surface of the earth and of the wall which supports
it are supposed to be both in one horizon tol plane.
1st. Both theory and experiment indicate that the resultant of the
thrust of a bank, behind a vertical wall, is at a distance B K from the
bottom of the wall equal to onethird of its height.
2dly. That the friction is half the pressure, in vegetoble earths,
and fourtenths in sand; or tliaty=s *5 in one case and '4 in the
other.
.11.]
PRBS8UBB OF BABTH AGAINST WALLS.
201
ly. The cohesion which vegetahle earths acquire, when cut in
and well laid, course hy course, diminishes their thrust hy full
kirdt; or in this case y = '667»
16 following tahle contains the value of the angle C B F, (heing
the angle formed hy the natural slope and the vertical,) for
al different kinds of earth, the authorities heing given in the last
on.
Nature of Earth.
Angle
Authority.
Inbble
Axme shingle perfectly dry
2arth the most dense and compact
.^ommon earth tliffhtlv damp .
Idem pulverized and dry
Vegetahle earth
Idem mixed with large g^vel
Idem mixed with imall g^vel
Sand
'Ine dry sand ....
Idem
Idem (a single experiment)
22 30
26 30
17 30
18
21 35
31 43
31 43
35 52
34 6
25 30
27 45
34 30
MaynieL
Pasley.
Barlow.
Rondelet.
Id.
MaynieL
Id.
Id.
Id.
Barlow.
Rondelet.
Oadroy.
THICKNESS OF WALLS, BOTH PACES VERTICAL.
The following tahle exhibits the thickness which ought to he
i to a parallel wall of various materials, and supporting the
ore of different kinds of earth.
Thickness of the wall or DC,
its height being 1.
Nature of Earth.
WaUof
brick.
100 Ibt.
pereub. ft.
WaU of un
hewn itooe,
lasibs.
per cab. ft.
Wall of
hewn
flreotooe.
taUe earth, carefully hud course by course
, weU rammed
li mixed with Uuge gravel
•16
17
•19
•33
•15
•16
•17
•30
•13
•14
•16
•26
. Far waUs with an inUricr slope^ or a slope towards the hank,
16 hose of the slope he — of the height, and let S and « he the
fie gravities of the wall and of the earth ; then
DC =^ A/ — I m ;
V 3»* S n
« m ^ •0424, for vegetahle or clayey earth, mixed with large
202 BQUILIBBIUM OF P0LT€K)1I8. [PABT n.
graTe] ; *0464>, if the earth be mixed with mall grmTel ; '1528, for
sand; and '166, for semifluid earths.
Example, Suppose the height of a wall to be 20 feet, and ^ of
the height for the base of the tcdm9 or slope; suppose, also, the
specific gravities of the wall and of the bank to be 2600, and 1400,
and the earth semifluid; what, then, must be the thickness of the
wall at the crown ?
Here the theorem will become.
^^='^Vi^+*^^^^^»
= 20 >/ 0008333 + 0894 — 1 = (20 x '3)  1
= 6 — 1 = 5 feet :
while the thickness of the wall at bottom will be 6 feeL
EQUILIBRIUM OF POLYGONS.
1. Let there be any number of lines, bars, or beams, AB, BC,
CD, DE, &c. (fig. 175), all in the same vertical plane, connected
together and freely moveable about the joints or angles. A, B, C, D,
£, &c., and kept in equilibno by weights laid on the angles : it is
required to assign the proportion of those weights ; as also the force
or push in the direction of the said lines ; and the horizontal thrust
at every angle.
Through any point, as D, draw a vertical line a J} fig; to which,
from any point, as C, draw lines in the direction of, or parallel to,
the given lines or beams, viz., C a parallel to A B, Ch parallel to BC,
Ce to DE, C/to EF, Cy to F O, &c. ; also CH parallel to the
horizon, or perpendicular to the vertical line adg^ in which also
all these parallels terminate.
Then will all these lines be exactly proportional to the forces
acting or exerted in the directions to which they are parallel, whether
vertioil, honxontal, or oblique. That is, the oblique forces or thrusts
in the direction of the bars AB, BC, CD, DE, £ F, F6,
are proportional to their parallels ... Ca, C6, C</, C^, CJ\ Cg;
and the vertical weights on the angles B, C, D, E, F, 9ic.
are as the parts of the vertical a 5, &D, De, ej\ fg^
and Uie weight of the whole frame ABCDEFQ,
is proportional to the sum of all the verticals, or to ag\
also the horizontal thrust at every angle, is everywhere the same
constant quantity, and is expressed by the constant horizontal
line CH.
Cord, 1. It is worthy of remark that the lengths of the bars A B,
B C, &c. do not affect or alter the proportions of any of these loads
or thrusts ; since all the lines C a, C ^, a 6, &c., remun the same,
whatever be the lengths of A B, B C, &c. The positions of the bars,
and the weights on the angles depending mutually on each other, as
well as the horizontal and oblique thrusts. Thus, if di^re be givea
CBAP. n.] BQUILIBBIUM OF P0LT001V1. 203
the positioo of D C, and the weights or loads laid on the angles D,
C« B; set these on the Tertical, DH, D6, ba^ then C&, Ca, give
the direetiooB or positions of C B, B A, as well as the quantity or
proportion of C H the constant horizontal thrnst.
Vorol. S. If C H be made radios ; then it is evident that Ha is the
tngent, aad C« the secant of the elevation of C a or A B ahove the
horiaon ; also H 6 is %he tangent, md C b the secant of the elevation
of C^ or CB ; also H D and CD the tangent and secant of the eleva
tion of C D ; also H e and C s the tangent and secant of the devadon
of C« or D B ; also B/ and C/ the tangent and secant of the eleva
tion of EF; and so on; also the parts of the vertical a&, 6D, e/, /y,
denoting the weights laid on the several angles, are the differences of
the said tangents of elevations. Hence then in general,
1st. The oblique thmsts, in the directions of the bare, are to one
another, directly in proportion to the secants of their angles with the
borixon; or, which is the same thing, reciprocally proportional to
the eosines of the same angles, or reciprocally proportional to the
sines of the vertical angles, a, ^, D, e, /^ y, &c., made by the vertical
line with the several directions of the bars; because (formula K . 3,
page 137) the secants of any angles are always reciprocally in pro
portion to their cosines.
2. The weight or load laid on each angle is directly proportional
to the difference between the tangents of the elevations above the
boriaon, of the two lines vrhich form the angle.
3. The horiaontal thrust at every angle is the same constant quan
tity, and has the same proportion to Uie weight on the top of the
q ppe raio st bar, as radius has to the taneent of the elevation of that
W. Or, as the whole vertical a^ is to the line C H, so is the weight
of the whole assemblage of bars, to the horizontal thrust.
4. It may hence be deduced also, that the weight or pressure laid
on any angle, is directly proportional to the continual product of the
sine of that angle and of the secants of the elevations of the bare or
lines which form it.
Sekokum. This proposition is very fruitful in its practical conse
quences, and contains the whole theory of centerings, and indeed of
arches, which may be deduced from the premises by supposing the
constituting bare to become very short, like arch stones, so as to fonn
the curve of an arch. It appeara too, that the horizontal thrust,
which is constant or uniformly the same throughout, is a proper
memsuring unit, by means of which to estimate the other thrusts and
pressures, as they are all determinable from it and the given posi
tions ; and the value of it, as appeare above, may be easily computed
from the uppermost or vertical part alone, or from the whole assem
blage together, or from t^y part of the whole, counted from the
top downwards.
In the most Important cases, a model of the structure may be
made, and the relations of the pressures at any angle, whether hori
zontal, ^erticsl, or in Qie directions of the beams, may be determined
by a spring steelyard applied successively in the several directions.
204 BQUILIBRIUM OP POLYGONS. [PABT II.
2. If the whole figure in the preceding prohlem be inverted, or
turned round the horizontal line AG (fig. 176) as an axis, till it be
completely reversed, or in the same vertical plane below the first
position, each angle D, dy &c., being in the same plumb line; and if
weights t, k^ /, m, n, which are respectively equal to the weights laid
on the angles B, C, D, E, F, of the first figure, be now suspended by
threads from the corresponding angles h^ e^ d^ d^ f^ of the lower
figure; those weights keep this figure in exact equilibrio, the same
as the former, and all the tensions or forces in the latter case,
whether vertical, horizontal, or oblique, will be exactly equal to
the corresponding forces of weight, pressure, or thrust in the like
directions of the first figure.
This, again, is a proposition most fertile in its application, especi
ally to the practical mechanic, saving the labour of tedious calcula
tions, but making the results of experiment equally accurate. It
may thus be applied to the practical determination of arches for
bridges, with any proposed roadway ; and to that of the position of
the rafters in a curb or mansard roof.
3. Thus, suppose it were required to make such a roof, with a
given width AE {^g, 177), and of four proposed rafters AB, BC,
CD, DE. Here, take four pieces that are equal or in the same
given proportions as those proposed, and connect them closely toge
ther at the joints A, B, C, D, E, by pins or strings, so as to be freely
moveable about them; then suspend the whole from two pins, A, E,
fixed in the same horizontal line, and the several pieces will arrange
themselves in such a form, A^c^/E, that all its parts will come to
rest in eauilibrio. Then, by inverting the figure, it will exhibit the
form of the framing of a curb roof A B C D E, which will also be in
equilibrio, the thrusts of the pieces now balancing each other, in the
same manner as was done by the mutual tensions of the hanging
festoon A6c</£.
4. If the mansard be constituted of four equal rafters; then, if
angle CAE = m, angle CAB=:;i?; it is demonstrable that 2 sin
2 ;r = sin 2 m. So that if the span A E, and height M C, be given,
it will be easy to compute the lengths AB, BC, &c.
EwampU. Suppose A E = 24 feet, M C 12.
IMF C*
Then, —— = 1 = tan 45° angle C A M = m.
MA
.'. sin 2 ??» 1= sin 90° = 1, and sin 2 ;r = ^
.. 2a; = 30°, and « = 15° =i CAB.
Hence M A B = 45° f 15° = 60°
and M B A = J (180° — 2 x 15°) = 90° — 15° = 75°
also A M B = 180° — (75° h 60°) = 45°
and lastly, sin 75° : sin 45 : : AM = 12 : AB = 8*7846 feet
Note. — In this example, since AM = MC, as well as AB = BC,
it is evident that AI B bisects the right anele A M C ; yet it seemed
nreferable to trace the steps of a general solution.
CHAP. II,] STABILITY OF ABCHBS. 205
STABILITY OF ABCHBS.
1 . If the effect of the force of gravity upon the ponderating matter
of an arch and pier, he considered apart from the operation of the
cements which unite the stones, &c., the investigation is difficult to
practical men, and it furnishes results that require much skill and
care in their application. But, in an arch whose component parts are
united with a veir powerful cement, those parts do not give way in
vertical columns, hut hy the separation of the ientire mass (including
arches and piers), into three, or, at most, into four parts ; and in this
case the conditions of equilibrium are easily expressed and applied.
LetyF, /'F', (fig. 178) be the joints of rupture, or places at
which the arch would most naturally separate, whether it yield in
two pieces or in one. Let G be the centre of gravity of the semi
arch /F K *, and G' that of the pier A B F/. Let 'f I be drawn
parallel to the horizon, and G H be demitted perpendicularly upon
It ; also let G^ D be a perpendicular passing through G^, and F £
diBwn from F parallel to it. Then,
2. The first case is when the arch fY Y' f tends to fall vertically
in one piece, removing the sections J^F, f'Y'\ if W be the weight
of the semiarch fY K k^ and P that of the pier up to the joint y F,
the equilibrium will be determined by these two equations : — viz.,
* • P = w 0?  *) (I.)
."=(n^0 ("■'
where ^ is the measure of the friction, or the tangent of the angle of
repose of the material, and the first equation is that of the equili
Imnm of the horizontal thrusts, while the second indicates the equili
lirium of rotation about the exterior angle A of the pier.
3. In the second case, when each of the two semiarches Yk^k F\
tend to turn about the vertex k of the arch, removing the points
F, F', the equilibrium of horizontal translation, and of rotation, will
he respectively determined by the following equations : — viz.,
* . p = w (j^i) (HI.)
'HC^ID (,
4. Hence it will be easy to examine the stability of any cemented
arch, upon the hypothesis of these two propositions. Assume dif
ferent points, such as F in the arch, for which let the numerical
values of the equations (I.) and (11.), or (III.) and (IV.) be com
puted. To ensure stability, the first members of the respective
eaoations must exceed the second ; and those parts will be weakest
where the excess is least
A
F£
206
ARCHBS AND PIBB8.
[part
5. The following table extracted from Prof. Moseley's ^^ Mechanical
Principles of Engineering," page 151, contains the value of the angle
^ for the materials most usually employed in the construction of
arches.
Nature of Materials.
GoeffidcBt
of
firiction.
Lumiting
Soft calcareous stone, well dressed, upon the same .
Hard calcareous stone, ditto
rommon brick, ditto
Oak, endwise, ditto
Wrought iron, ditto
Hard calcareous stone, well dressed, upon hard cal I
careous stone I
8oft, ditto
Conamon brick^ ditto
Oak, endwise, ditto
Wrought iron, ditto
Soft calcareous stone upon soft calcareous stone, with 1
fresh mortar of fine sand )
EZPERTUENTS BT DIFFERENT OBSERVERS.
Smooth freestone upon smooth freestone, dry. >
(Rennie.) y
Ditto, with fresh mortar. (Rennie.)
Hard polished calcareous stone upon hard polished }
calcareous stone • . . . . S
Calcareous stone upon ditto, both surfaces being made \
rough with the chisel. (Bouchard!.) . . >
Well dressed granite upon rough granite. (Rennie.)
Ditto, with fresh mortar, ditto. (Rennie.)
Box of wood upon pavement. (Ili^ier.)
Ditto, upon {>eaten earth. (Herbert.)
Libage stone upon a bed of dry clay
Ditto, the clay being damp and soft
Ditto, the clay being equally damp, but covered with 7
thick sand. (Oreve.) )
74
•76
•67
•63
•49
•70
•75
•67
•64
•42
•74
•71
•68
•78
•66
•49
•68
•33
•61
•34
•40
36 30
36 62
33 50
32 13
26
35
36 62
33 60
32 37
22 47
36 SO
36 23
33 26
30 7
37 68
33 26
26 7
80 7
18 16
27 2
18 47
21 48
If the section be drawn on smooth drawing pastehoard, upon a
good sized scale, the places of the centres of gravity may he found
experimentally, as well as the relative weights of the semiarch and
piers, and the measures of the several lines from the scale employed
in the construction.
If the dimensions of the arch were given, and the thickness of the
pier required ; the same equations would serve ; and different thick
nesses of the pier might he assumed, until the first members of the
equations come out largest.
The same rules are applicable to domes, simply taking the un
gulas* instead of t^e profiles.
* The ungulas^ as mentioned above, are the solids generated by the revolu
tion of the sections /FirK and /FA B about the vertical axis ArC.
CflAP. II.] 8U8PENM0II BBIDOBB. SOT
BQUILIBBIUH OF SUSPENSION BRIDGES.
1. The several parts of a suspension bridge should always be pro
portioned in such a manner, that the tensile strain per square inch of
section would be nnifbrm throughout the whole length of the chains ;
and as the tension varies with the inclination of the chain, and to
fulfil the above condition, the area of the chain must vary as the
tension, it is q«ite evident that the chains of a suspension bridge
ought not under any circumstances to be made of a uniform sectional
area.
If the chain was of a uniform section, and had only its own weight
to carry, the form which it would assume would be a catenary; if on
the other hand, the chain is supposed devoid of weight, and the load
suspended fhnn it to be uniformly distributed horixontally, (as in the
case of the roadway of a bridge,) then the form which the chain
would assume would be the common parabola. Neither, however, of
these supposed cases ever occur in practice, there being always three
loads very differently distributed, viz., the weight of the chain itself,
that of the roadway suspended from it, and that of the vertical rods
by which the same is suspended ; consequently, the form which the
chain when in equilibrium would assume is neither a catenary, nor
a common parabola, but is between the two; and it approaches
nearer to one or the other, as the weight of the chain or that of the
roadvray predominates. Taking these three loads into accoun^ and
varying the section of the chain so that its tensile strain is uniform
tbroashout, the determination of the true form of the curve which it
vodd assume when all its parts were in equilibrium, becomes a very
eoiDplicated problem; it Ims, however, been very ably solved by
Professor Moseley*, and the following formula which he has de
daoed, contains aU that is required for determinmg the form of the
eorre end the dimensions of the chains.
If fi^ = the weight of a bar of the material of the chain 1 square
inch in section and 1 foot long, /a, :s the weight of a foot in length
of the roadway, and, supposing the vertical suspending rods to be
diflfbsed over the whole space between the chain and the road
]Bray, formins a uniform piate of such a thickness that its weight
is precisely uie same as that of the actual suspending rods, let ^13 =
the "weight of a square foot of this plate ; also, let K := the sectional
%Tea in square inches of the chain at any point P (fig. 179), x and y
being the ordinate and abscissa at the same point; let a = the
Bemispan, H = the deflexion, b = the length of the shortest sus
pending rod, € =3 the tension upon the lowest point of the chain,
mMkd m c3 — ^^ where r represents the tenacity of the material of
T
Che chain per square inch, and m = the number of times that r
exceeds the actual tensile strain upon the chains ; then
* ** The Mechanical Principles of Engineering and Architecture,*' by the
Her. Henry Af oaeley, page 647*
208 SUSPENSION BBID0B8. [PART II.
 = C.>e'.Uj »') <■•)
"(iriir^y ("•)
K _^^2(,_ j)(!ii±!U + .) + ,}'.. (II,.)
2. As, however, the use of these formulsB for determining the
requisite numher of points in a hridge would be attended with much
labour, it is only in cases where great accuracy is requisite that they
need be employed ; for most purposes it will be sufficiently correct
to assume that the form of the curve is a common parabola, in which
case the formulse for determining the several elements of the bridge
become much more simple, and easy of application.
Let t ^ the coefficient of tension at the lowest point of the chain,
or the quantity by which half the weight of the bridge must be
multiplied to give the actual strain upon the chains, t^ = the coeffi
cient at the point of suspension, t^ := the mean coefficient of tension
for the whole chain, « := the semispan, d = the deflexion, x =
the leoeth of half the chain, K^ = the sectional area of the chains
at the lowest point, K^ = the same at the point of suspension,
K3 := the mean sectional area of the whole chain, W := the weight
of half the whole bridge, including the chain, y = the angle which
the tangent to the curve at the point of suspension makes with the
horizon, and b^ = the length of the suspending rod at any point P.
The letters, /a^, /a^,, m, t, ^, :r, ^, and c, represent the same quan
tities as before.
Then, the curve being a parabola, the length of the suspending rod
at any point P may easily be found by means of the following
property of the parabola ;
y : rf : : a?' : «,
therefore, supposing the roadway to be horizontal, we have
h, =^ + V^ (IV.)
And for the length of the chain, we have from the formula at
page 129, for the length of parabolic arcs.
=V
•* + rf' (V.)
Also, from the method given at page 174, for drawing a tangent
to any point in a parabola, we can easily deduce the value of 7, as
follows: — if DE (fig. 180) be made equal to C D, the line drawn
from A to £ will be the tangent fo the parabola at the point A,
and therefore the angle CAE = 7, is determined from the proportion
B : rad : : 2d : tan 7;
CHAf. II.] SUSPENSION BRIDGES. 209
.. tany = — (VI.)
^iid sioce the chain A D B with the roadway suspended from it
^ notbiog more than an equilibriated polygon, such as is described
at page 202, having an infinite number of sides, all the relations
there mentioned as existing between the several strains, may be
applied to the present inquiry; therefore, if CE = 2rf be made to
"^Pwsent the whole weight of half the bridge = W, then A C s= «
^^^ represent the tension at D, and A E the tension at A ; therefore,
''=2^ (V")
^ = '^ (VIII.)
•nd <, = (I + «sc>) ^ (IX.)
lien, for the mean sectional area of the chain, we have
K,= ^^»' (X.)
•"^d for the weight of the roadway and chain for half the bridge,
W = K3X^j+fi2« (XL)
*Tjen since /j W = c, the tension at the lowest part of the chain,
^^ have for the sectional area of the chain at the same place,
K, =^!lZ (XII.)
T
^^ K W being the tension at the point of suspension, the sectional
**^ of the chain at that point will be
K, =^^ ^Xjjj^
^ ^* In order to render the practical application of the foregoing
^i^nle quite clear, we subjoin the following example. Let it be
'^iriTed to determine all the elements of a suspension bridge, tlie
•P*Ji of which is to be 360 feet, and the deflexion 30 feet; let the
y^^} of the roadway for everj' foot in length = 4500 lbs., the
^^**ciiy of a square inch of the chain = 67200 lbs., the weight of a
^ 1 inch square and 1 foot long := 3*4 lbs., and the number of
r^ that the cohesive strength of the iron is to exceed the
"^ mbstituting these values for the letters representing the
^«il qumiities, we obtain from formula (V.)
p
210 SUSPENSION BRIDGES. [Pi
4 X SO'*
180 + — = 18333 feet, equal the length c
the chain.
From formula (VI.)
2 X 30 1
tan V = =  = 3333, which is the tangent* of li
'^ 180 3 ' ^
the angle made hy the chains with the horizon at their pc
suspension.
From formula (VII.)
180
t, = = 3, the coefficient of tension at the lowest i
* 2 X 30
the chain; from formula (VIII.)
180 X 10541 1 „,^^„ , ,, «• * r*
/ = = 3*1623 equal the coefficient of teui
2 2 X 30 ^
the point of suspension ; and from formula (IX.)
180
f, =(1 f 10541) . =30812 for the mean coeffici
tension for the whole length of the chain.
From formula (X.)
4500 X 180
67200
— 1— —   18333 X 34
6 X 30812
mean sectional area of the chain.
From formula (XL)
W = 269 X 18333 x 3;4 + 4500 X 180 = 977,647 lbs. \
total weight of half the bridge; and finally, from formula (XI
obtain
6 X 3 X 977,674 ^^^ • u r *a.
Kj = =262 square mches for the section
of the chain in the centre ; and from formula (XIII.)
ex 31623 X 977,674 ^^^ • t, r .u
Kp = ^^^^^ = 276 square inches for the se
* 67,200 ^
areA of the chain at the point of suspension.
"• In order to find the angle of which '3333 ii the tangent, take tl
'3333 a 7*522835, and adding 10 to the characteristic (for the reason <
which see page 153), we have 9*522835, which we find from Table IV
log tan of IS"" 26^, as above.
t The secant of 18** 26' is obtained by a reverse process to that ezpl
the foregoing note ; thus, from Table IV. we find the log sec of W\
10*022875, then subtracting 10 we have 0*022875, the number answ
which in Table III. is 1 0541, the secant of 18** 26^ radius being 1.
K3 = cpfa/\t\ ' — *^~ ^ ^^^ square inches f
CHAP, in.] GBNBRAL DBFINITIONS. 211
CHAP. III.
DYNAMICS.
Sect. I. General Dejtnitions,
1. The man of a body is the q^uantity of matter of which it ift
«wnp08ed; and is proportional to its weight, or to the^brc^ which
nost be applied to the body to prevent its gravitating to the earth,
i&d wbicb, being greater or less as the mass is greater or less, we
i^u a measure of the mass itself.
3. Demit^ is a word by which we indicate the comparative close
^ or otherwise of the particles of bodies, and is synonymous with
^ term tpeeifie gravity. Those bodies which have the greatest num
^ of particles, or the greatest quantity of matter, in a given magni
^C) we call moA dense; those which have the least quantity of
^*il(tt, leatt dejtse. Thus lead is more dense than freestone ; yre«
^^ more dense than oak ; and oak more dense than cork.
. 3. The velocity with which a body in motion moves, is measured
^y the space over which it passes in any given time; the unit usually
^^Mned being one second.
4. If the body passes over an equal space in each successive unit
^ time, the body is said to move uniformly^ br to have a uniform
^^ki^y and the measure of such velocity is the space actually passed
^^er b? the body in each second.
S* If, however, the body passes over a yreafer space in each sue
^^▼e second tlwn it did in the preceding, then it is said to move
^tt ao aeceleraied vdocUy; when the differences between the spaces
^^^ oter in any two successive seconds is the same, at whatever
l^^nod of the body's motion they be taken, or in other words, when
^ HKceaaive spaces form an arithmetical progression, the body is
^^ to move with a uniformly accelerated velocity ; but when the
^itts passed over in successive seconds increase according to any
^^ law, the body is then said to have its velocity variably accele
^ If^ on the other hand, the body passes over a smaller space in
^ SQoeeasive second than it did in the preceding, then it is said to
^e with a retarded velocity; which, if the successive spaces form a
^<c<eaaiig arithmetical series, is said to be uniformly retarded; if
^^kerwiae, it is said to be variably retarded,
p 2
212 LAWS OF UNIFORM AND VARIABLE MOTION. [PART II.
7. The velocity of a body whose motion is Tariable is expressed at
any moment, by the space which it tcould pa^s ocer in a second^ if its
Telocity at the moment spoken of were to continue uniform for that
period.
8. Mechanical effect is measured by the product of the nuus or
weight of the body into the space over which it has been moved; no
regard being had to the time occupied. The unit of mechanical effect
employed in the subsequent pages, is a weight of 1 pound raised
through a space of 1 foot, and is designated by the letter U.
0. The momentum of a body in motion means the mechanical
effect which such a body will produce in a moment (or second) of
time, and varies as the weight of tlie body multiplied by its velocity,
10. The vis viva of a body in motion is the whole mechanical
effect which it will produce in being brought to a state of rest^ no regard
being had to the time in which the effect is produced, and it varies as
the weight of the body multiplied by the square of its vel4>city.
Sect. II. On the general Latcs of Uniform and Variable Motion.
1. As a proper understanding of the actual difference between the
momentum and the vis viva of a body in motion, is esjtential to a
correct application of the principles of dynamics, we shall take some
pains to set this difference in as clear a light (for the student) as pos
sible. And it is of the more importance to do so, ns a diversity of
opinion upon this subject has existed amongst some of the most emi
nent mathematicians, and much time and talent has been expended
by them in supporting errors w Inch have arisen entirely in a miscon
ception of terms, ai)d in excluding from their conclusion the con
sideration of time which they included in their premises.
2. From carefully conducted and often repeated experiments, the
following results with regard to bodies in motion have been ob
tained : —
I. If a body of a certain weight, and moving with a given
velocity, meet another body of double that weight, and
moving with half the velocity, the two bodies will destroy
each other's motion, and both will be brought to a state of
rest.
II. A body of a certain weight and moving with a given velo
city, being subject to a uniformly retarding forc^, (i . e, a uni
form force acting constantly in a contrary direction to the
body's motion,) will move over a certain space in being
brought to rest, and will occupy a certain time in doing so ;
then another body of the same weight, but moving with half
the velocity of the former, being Subject to the same uni
formly retarding force, will move over one quarter of the
CHAP. III.] LAWS OF UNIFORM AND VARIABLB MOTION. 213
space moTed over by the former, in being brought to a state
of rest, and will occupy in doing so ^o^tbe time. And an
other body of the same weight, but moving with onethird of
the velocity of the first, will move over oneninth of the space,
and occupy onethird the time of the first, in being brought to
a state of rest.
Now the diversity of opinion to which we have alluded above, has
arisen from the (at first sight) apparently contradictory nature of
these two results : one party has drawn a conclusion from the first
experiment that the force of a body in ynotion is directly <w its velo
city; and the other party has drawn a conclusion from the second
experiment that \he force of a body in motion is directly as the square
of its velocity. These errors have arisep from the term " force of a
body in motion" being used without any fixed and definite meaning
being attached to the same.
3. The proper measure of the whole force (i. e. vis viva) of a body
in motion, is the mechanical effect, or (as Professor Moseley very
significantly terms it) the work^ which it is capable of performing in
being brought to a state of rest. That evidently being the force due
to its faction which is required to destroy the same^ and which is
directly as the square of its velocity^ as found by the second experi
ment.
4. If, however, we only consider the mechanical effect (or the
work) which a body in motion is capable of performing in a given
time, (i.e. its momentum^) we shall find from the second experiment,
that ajthough the body having twice the velocity ultimately produced
four times Qie effect, in doing so it occupied just twice the tii>c; and
^in, although the body moving with three times the velocity tUti
niatefy produced nine times the effect, it occupied in doing so three
times the time ; and, therefore, the mechanical effect produced in a
yicen time by the bodies was directly as tlie velocities. And this con
clusion, drawn from the second experiment, is in accordance with
that which must be drawn from the first experiment, in which it is
eyident that the effect produced by both bodies must be in the same
(although indefinitely small) time.
5. The force of gravity being constantly the same, both in amount
%iid direction, and being practically uniform* in its action for such
^niall distances from the earth's surface as come under consideration
iQ ordinary dynamical investigations, has been universally adopted as
the onit of measure for all other forces.
6. The actual amount of the force of gravity is measured by the
Velocity which a body falling in vacuo, in the latitude of London, will
* The force of gravity varies as the square of the distance from the earth*s
oencre, and therefore becomes less as we asueiid alxive the surface of the earth,
in the proportion of I to *9994, a difference too small to require notice in any
question of terrestrial mechanics.
214 MOTION UNIFORMLY ACCBLBBATED. [PABT II.
acqaiie in one second of time, and which, hy carefully conducted ex
peiiments, has heen ascertained to he 386*289 inches, or about 32^
feet per second^ and this latter quantity is usually represented by the
symbol g.
MOTION UNIFORMLY ACCBLERATED.
1 . If we now put W to denote the quantity of matter, or weigLt
of a body in motion, v its velocity in feet per second, M its momen
tum, and V its vU viva^ both expressed in units of mechanical effect,
or pounds raised through a space of 1 foot; we have
M = Wfj (I.)
^ = ^2^ <"•>
2. And further, if s h? put to represent the space passed over in
the time ^, by a body subject to the uniform forced*; then we have
the following relations between all these quantities : —
V = «W = W=yW = 2/W . (III.)
2 22/ ^^'>
«=//=—= ^2fB .... (VI.)
' = ) = va/7 •• <™>
MOTION OF BODIES UNDER THE ACTION OF GRAVITY.
1. When the uniform force is that of gravity,/ := g; and g being
proportional to the weight of the body (W), the foregoing relations
are simplified, and then become as follows : —
, = ^ = ^J! = ^ (IX.)
2 2 2^ ^ ^
V^gt^— ^^/Yf% (X.)
CHAP. III.] DE8CBNTS BY OBAVITV. 215
= • = — = /u
(XI.)
=r7' = n '■)
2. Any two of these quantities being given, the other two may be
immediately ascertained from the above equations; the following
table shows their actual numerical values for the first four seconds of
the motion of a heavy body filling freely by the action of gravity : —
The times in seconds being 1", 2", 3", 4", &c.
The velocities in feet will be 32^, 64 j, 96i, 128^, &c.
The spaces in the whole times l6j^^ 64, 144}, 257, &c.
And the space for each second 16^, 48^, 80j^, 112^*^, &c.
of which spaces the common difference is 32^ feet, equal y, the mea
sure of the force of gravity.
3. If, instead of a heavy bodv being allowed to fall freely, it be
propelled vertically upwards or downwards with a given velocity, v,
then
M^tvzfi^; (XIII.)
aa ezpressioD in which the upper sign — must be taken when the
projection is upwards^ the lower sign + when the projection is down
wardM.
4. When only an approximate result is required with reference to
bodies falling vertically, 32 may be put for ^, instead of 32 J : there
would then result, in motions from quiescence,
''"■a"? • ■ • ■ (»"'•)
t;=8v^i=: — = 32^ (XVI.)
^lius, if the space descended were 64 feet, we should have i; := 8
Q
^ B = 64 feet per second, and ^ =  = 3 seconds.
If the space descended were 400; then t? = 8 x 20 = 160 feet
l^,e«o.d,«.d. = ¥ = 6eeco»dB.
4
5 The force of gravity differs a little at different latitiules ; the law
^ the variation is not as yet precUely ascertained ; but the following
216 MOTION OVEB PULLEYS, ETC. [PART II.
theorems are known to represent it very nearly. That is, if ^ denote
the force of grayity at latitude 46% ^j the force at the poles, ^^ the
force at the equator, and ^3 the force at any other place : then
^, =^(1 + 002837) (XVII.)
g^z=g{\ — 002837) (XVIII.)
g^^g (1 — 002837 cos 2 lat.) . . (XIX.)
MOTION OVER A FIXED PULLEY.
In this case let the two weights which are connected hy the cord
that goes over the pulley be denoted by W . and W , : then — ^ — — ^ 9
= W/in formula (III.); so that
W,W, gj_
W, +W, ' 2
Or, if the resistance caused by the rigidity of the rope, and the fric
tion and inertia of the pulley, be represented by r; then
'W,+W, + r • 2 ^^^^
Example 1. Suppose the two weights to be 6 and 3 lbs. te&Y*^^'
ively, what will be the space descended in 4 seconds ?
16 = 16jV X 4 = 64jfeet.
Example 2. But suppose that, in an actual experiment witl» ^
weights of 6 and 3 lbs. over a pulley, the heavier weight desce^^ ^^
only 50 feet in 4 seconds.
the same in both examples,
we have w, + W^ + r : W. + W, :: 64J : 60
or, dividendo r : W, f W^ :: 14J *: 60
that is, r : 5 f 3 :: 14J : 60
whence r = ^^tJ^ = ^4^^ = 22933 lbs.
50 50
the measure of the resistance and the inertia.
2. Similar principles are applicable in a variety of other caies /
and by varying the relations of W„ W^, and r, the force may hvre
CHAP. III.] MOTION ON INCLINED PLANES. 21?
any assigned ratio to that of gravity; which is, indeed, the founda
tion of Mr. Atwood's elegant apparatus for experiments on accele
rating forces; an account of which may he seen in the 2nd volume of
my Mechanics^ or in almost any of the general dictionaries of arts
and sciences.
3. If, instead of pulleys, small wheels and axles, as in figure 181,
be employed^ to raise weights by the preponderance of equal weights:
then, if the diameter of the wheel and axle A be as 3 to 2 ; those of
the wheel and axle B, as 5 to 2; and those of C, as 8 to 2; it will
be found that the weight b will be elevated more rapidly than either a
or c : the proportion of 5 to 2, (or, more accurately, of 1 f ^^2 to I)
being in that respect the most favourable.
MOTION ON INCLINED PLANES.
1 . When bodies move down inclined planes, the accelerating force
(independently of the modification occasioned by the position of the
centre of gyration) is equal to g multiplied by the quotient of the
beight of the plane divided by its length, or what is equivalent, by
the sine of the inclination of the plane. In this case, therefore, put
ting « for the inclination of the plane, the formulae become
(XXI.)
_ ^ ^ . sin I ^ tv
2 2^ . sin f "" 2
2 9
r = y / . sin I = >/ {^g9 . sin t) = — (XXII.)
*=./^'=^' (xxm.)
Purther, if », be the velocity with which a body is projected up or
"Oivn a plane, then
c = Vj ip y / . sin I (XXIV.)
^^ . sin f v^ — v^ ,
s=^v^t^^ =~^ —. . . (XXV.)
* ^ 2 2^ . siuf ^ ^
Alaking v, = 0. in equation (XXIV.), and the latter' member of
^^ Nation (XXV.), the first vnll give the lime at which the body will
'^^'^^e to rise, the latter the spctce.
Example. Suppose a body be projected up a smooth inclined
p^Be whose height is 12 and length 193 feet, with a velocity of 20
.^^t per second, how high will it rise up the plane before its motion
^^ extinguished?
Then substituting these values in formula (^XXV.), wc have
4000 400 400 ,_^ , . ,
"= 64J X ^»/, =193 12 =^ = 100fcet,thcspacerequired.
3 ^ 193
218 MOTION ON INCLINED PLANES. [PART II.
2. With regard to the velocities acquired by bodies in falling down
planes of the same height, this proposition holds ; viz. that they are
all eaual, estimated in their respective directions. Thus, if AD, BE,
CF, (fig. 182), be planes of different inclinations, and AC, DF, hori*
zontal lines, the balls A, B, C, after descending along those planes
will have equal velocities when they arrive at the points D, £, F, re
spectively.
3. Also, all the chords, such as AD, BD, CD, (fig. 183), that ter
minate either in the upper or the lower extremity of the vertical dia
meter of a circle, will be described in the same time by heavy bodies
A, B, C, running down them; and that time will be equal to die time
of vertical descent through the diameter DE.
4. If three weights, asA, B, C, (fig. 184), be drawn up three
planes of different inclinations, by three equal weights hanging from
cords over pulleys at P, then if the length of the middle plane be
ti0ice its height^ tLe body B will be drawn up that plane, quicker than
either of the other weights A or C. Or, generally, to ensure an ascent
up a plane in the least time, the length of the plane must be to its
heighty as ttoice the weight to the power employed.
5. If it be proposed to construct a roof over a building of a given
width, so that the rain shall run quickest off it, then each side of the
roof must be inclined 45° to the horizon, or the angle at the ridge
must be a right angle.
6. The force by which spheres, cylinders, &c. are caused to re
volve as they move down an inclined plane (instead of sliding) is the
adhesion of their surfaces occasioned by the pressure against the
plane : this pressure is part of the body's weight ; for the weight
being resolved into its components, one in the direction of the plane,
and the other perpendicular to it, the latter is the force of the pres
sure upon the plane; and, while the same body rolls down the plane,
will be expressed by the cosine of the plane's elevation. Hence,
since the cosine decreases while the arc or angle increases, after the
angle of elevation ariives at a certain magnitude, the adhesion may
become less than what is necessary to make the circumference of the
body revolve fast enough ; in this case the body descends partly by
sliding and partly by rolling. And the same may happen in smaller
elevations, if the body and plane are very smooth. But at all eleva
tions the body may be made to roll by Uie uncoiling of a thread or
riband wound about it.
If W denote the weight of a body, a the space described by a body
falling freely by the action of gravity, or sliding freely down an in
clined plane, then the spaces («,) described by rotation in the same
time by the following bodies, will be in these proportions.
(1.) In a hollow cylinder, or cylindrical surface, « = ^ « and the
tension of the cord, in the first case =  W.
(2.) In a solid cylinder, «, = ^ «, and the tension = ^ W.
(3.) In a spheric surface, or thin spherical shell, «, = ^ «, and the
tension =  W.
(4.) In a solid sphere, « =: ^ «, and the tension = ^ W.
CHAP, in.] MOTION ON INCLINBD PLANES. 210
If two cylinders be taken of equal size and weight, and with equal
protuberances upon which to roll, as in the mar
ginal figures: then, if lead be coiled uniformly
over the curve surface of B, and an equal quan
tity of lead be placed uniformly from one* end
to the other near the axis in the cylinder A, that
cylinder will roll down any inclined plane quicker
than the other cylinder B. The reason is that
each partide of matter in a roUing body reiisU
motion in proportion to the squarb of its distance
JrofH ike axis of motion ; and the particles of lead
which most resist motion are placed at a greater distance from the
axis in the cylinder B than in A.
7. The friction between the surface of any body and a plane, may
be eanly ascertained by gradually elevating the plane untiJ the body
upon it/iM/ begins to dide. The friction of the body is to its weight
as the height of the plane is to its base, or as the tangent of the in
clination of the plane is to the radius. Thus, if a piece of stone in
weight 8 pounds, just begins to slide when the height of the plane is
2 feet, and its baise 2\ ; then the friction will be ^ the weight, or 4
of 8 lbs. = 6 lbs.
8. After motion has commenced upon an inclined plane, the fric
tion is osoally much diminished. It may easily be ascertained expe
Timentaiiy, by comparing the dme occupied by a body in sliding down
a plane of given height and length, or given inclmation, with that
which the simple theorem for /, (XXIII.) would give. For, iff he
the value of the friction in terms of the pressure, the theorem for
the time wiU be
^1 = A / — T' "^y instead off = a / : — • Hence
r,«:<«::sini:sini/ . . . (XXVI.)
Example. Suppose that a body slides down a plane in length 30
feet, height 10, in 2^ seconds, what is the amount of the friction?
Here f « A / ^ = A /;;7n r = 2366 nearly.
Hence (26)» : (2366)' :: § : 27603 = sin • /
Consequently, 33333 — 27603 = 0573 value of the friction, the
weight being unity.
9. When a weight is to be moved either up an inclined plane, or
along on horizontal plane, the angle of traction PWB (fig. 185) that
the weight may be drawn with least effort, will vary with the value
off. The magnitude of that angle PWB for several values of f are
exhibited in the following table: —
220
MOTIONS ABOUT A PIXBD CBNTBR OB AXIS. [PABT IT.
/
PWB
1/
PWB
/
PWB
/
PWB
/
PWB ij/
PWB
1
45° 0'
II
26°34'
^
18^26'
i
14° 2'
t
11°19! ^
9^28'
t
38 40
23 58
^
16 54
tV
13 15
i\
10 47 IJ4
8 8
^
33 41
^
21 48
f
15 57
^
12 32
A
10 18
4
7 8
+
29 45
A
19 59
h
14 56
A
11 63
A
9 52
i
6 20
10. If, instead of seeking the line of traction so that the moving
force should he a nainimuro, we required the position such that the
suspending force to keep a load from descending should he a mini
mum, or a given force should oppose motion with the greatest energy;
then the angles in the preceding tahle will he still applicahlc, only
the angle in any assigned case must he taken helow, as BWp. This
will serve in the huilding and securing walls, in the construction of
hanks of earth, fortifications, &c., and in arranging the position of
landties^ &c.
Sect. III. Motions abotU a fixed Center or Axis.
Centers of Oscillation and Percussion.
1. The center ofosciUation is that point in the axis of suspension
of a vihrating hody in which, if all the matter of the system were
collected, any force applied there would generate the same angular
velocity in a given time as the same force at the center of gravity, the
parts of the system revolving in their respective places.
Or, since the force of gravity upon the whole hody may be con
sidered as a single force (equivalent to the weight of the hody)
applied at its center of gravity, the center of oscillation is that point
in a vihrating hody in which, if the whole were concentrated and
attached to the same axis of motion, it would then vihrate in the
same time that the hody does in its natural state.
2. From the first definition it follows that the center of oscillation
is situated in a right line passing through the center of gravity, and
perpendicular to the axis of motion. It is always farther from the
point of suspension than the center of gravity.
3. The center of percussion is that point in a hody revolving about
an axis, at which, if it struck an immovable obstacle, the whole of
its motion would be destroyed, or it would not incline either way.
4. When an oscillating body vibrates with a given angular velo
city, and strikes an obstacle, the effect of the impact will be the
greatest if it be made at the center of percussion. For, in this case
the obstacle receives the whole revolving motion of the body;
whereas, if the blow be struck in any other point, a part of the
motion of the body will he employed in endeavouring to continue the
rotation.
5. If a body revolving on an axis strike an immovable obstacle
CHAP. III.] PENDULUMS. 221
at the center of percussion, the point of suspension will not he
affected hy the stroke. We can ascertain this property of the point
of suspension when we give a smart hlow with a stick. If we give
it a motion round the joint of the wrist only, and, holding it at one
extremity, strike smartly with a point considerahly nearer or more
remote than ^ of its length, we feel a painful wrench in the hand :
but if we strike with that point which is precisely at ^ of the length
(that being the situation of the center of percussion), no such dis
agreeable strain will be felt. If we strike the blow with one end of
the stick, we must make its center of motion at ^ of its length from
the other end; and then the wrench will be avoided.
6. The distance of the center of percussion from the axis of motion
is equal to the distance of the center of oscillation from the same :
supposing that the center of percussion is required in a plane passing
through the axis of motion and the center of gravity.
SIMPLE AND COMPOUND PENDULUMS.
1 . A simple pendulum^ theoretically considered, is a single weight,
regarded as a point, or as a very small globe hanging at the lower
extremity of an inflexible right line, void of weight, and suspended
from a fixed point or center, about which it oscillates.
2. A compound pendulum is one that consists of several weights
movable about one common center of motion, but so connected
together as to retain the same distance both from one another and
from the center about which they vibrate.
Or any body, as a cone, a cylinder, or of any shape, whether
regular or irregular, so suspended as to be capable of vibrating, may
be regarded as a compound pendulum ; and the distance of its center
of oscillation from any assumed point of suspension, is considered as
the length of an equivalent simple pendulum.
3. If O represent the distance of the center of oscillation from the
point of suspension, and G the distance of the center of gravity from
the same point, it has been found that 6 O is a constant quantity, for
the same body and the same plane of vibration.
4. Any such vibrating body will have as many cefiters of oscillation
as yon give it points of suspension: but when any one of those
centers of oscillation is determined, either by theory or experiment,
the rest are easily found by means of the foregoing property that O G
is a constant product, or of the same value for the same body.
5. When a body either revolves about an axis, or oscillates, the
sum of the products of each of the material elements, or particles of
that body, into the squares of their respective distances from the axis
of rotation, is called the momentum of inertia of that body.
6. A point, or very small body, on descending along the successive
sides of a polygon in a vertical plane, loses at each angle a part of
its actual velocity equal to the product of that velocity into the versed
sine of the angle made by the side which it has just quitted, and the
222 PENDULUMS. [part II.
prolongation of the side upon which it is just entering. Therefore,
that loss is indefinitely small in curves,
7. A heavy body which descends by the force of gravity along a
curve situated in a vertical plane, has, in anv point whatever, the
same velocity as it would have had if it had fallen through a vertical
line equal to that between the top and the bottom of the arc run
over : and when it has arrived at the bottom of any such curve, if
there be another branch either similar or dissimilar, rising on the
opposite side, the body will rise along that branch (apart from the
consideration of friction) until it has reached the horizontal plane
from which it first set out. Thus, after having descended from A to
V (fig. 186), it will have the same velocity as that acquired by
falling through D V, and it will ascend up the opposite branch until
it arrives at B.
8. If the body describe a curve by a pendulous motion, the same
property will be found to obtain, setting aside the effects of friction.
Thus, let a ball hang by a flexible cord SD (fig. 187) from a pin
at S : then, after it has descended through the arc DE, it will pass
through an equal and similar arc £A, going up to A in the same
horizontal line with D, and ascending from £ to A in an interval of
time equal to that which it descended from D to £. But, if a pin
projecting from P or p stop the cord in its course, the ball will still
rise to B or to C, in the same horizontal line with A and D ; but will
describe the ascending portions of the curve in shorter intervals of
time than the descending branch.
9. When a pendulum is drawn from its vertical position, it will be
accelerated in the direction of the tangent of the curve it would
describe, by a force which is as the sine of its angular distance from
the vertical position. Thus, the accelerating force at A (fig. 188),
would be to the accelerating force at B, as A F to BE. (See art. 5,
on the Center of Gravity,) This admits of an easy experimental
proof.
10. If the same pendulous body descend through different arcs, iu
velocity at the lowest point will be proportional to the chords of the
whole arcs described. Thus, the velocity at D, after passing through
ABD, will be to the velocity at D after descending through ^e
portion BD only, as AD to BD.
1 1 . Farther, the velocity after describing ABD, is to the velocity
after describing BD, as v^FD is to v^ED. If, therefore, we would
impart to a body a given velocity V, we have only to compute the
V* V*
height FD, such that FD = — = — r feet, and through the point
F draw the horizontal line FA; then, letting the body descend as a
pendulum through the arc ABD, when it arrives at D it will have
acquired the required velocity. This property is extremely useful in
experiments on the coUision of bodies.
CHAP. III.] PENDULUMS. 223
12. The oscillations of penclulums in any arcs of a cycloid are
ifockrofud^ or performed in equal times.
13. Oscillations in »n)aU portions of a circular arc are nearly
iiockronal,
14. The numbers of oscillations of two different pendulums, in
the same time, and at the same place, are in the inverse ratio of the
square roots of the length of those pendulums.
15. If / be the length of a single pendulum, or the distance from
the point of suspension to the center of oscillation in a compound
pendulum, g = the measure of the force of gravity (32^ feet, or
386 inches at the level of St. Paul's* in the latitude of London),
/ the time of one oscillation in an indefinitely small circular arc,
And w =s 3*141593 : then
=v: <■■'
16. Conformably with this we have
39 \ inches, length of the second \ pendulum
9Jf inches half second in the
4^ inches third of second latitude
2^^ inches quarter second) of London.
17. Putting y, for the force of gravity in any latitude and at any
^titude, we have also /= 20264 x ^
and 1^1 = 49348/ (IL)
In other words, whatever be the force of gravity, the length of a
second pendnlum, and the space descended freely by a falling body
in 1 second, are in a constant ratio,
18. If /, be the length of a pendulum, ^, the force of gravity,
and /, the time of oscillation at any other place, then
'• ■■■■ ^/i ■■ Vf. '■"
)
If the force of gravity be the same,
t :t,:: ^l: ^ I, (IV.)
If the aune pendulum be actuated by different gravitating forces,
we have
* At the level of ike mo, in the latitude of liondon, g is 386289 inches, and
the c o rre ^ w m dipg length of the necond penduhim is 39*1393 inches, according
to the determination of Major Kater. Conformably with this result are the
Biunbers in the Table (in the Appendix), computed at the expense of Messrs.
BrawnA and DoniUn, and obli^nngly communicated by them for this work.
It has been suspected by M. Beuel'^ and demonstrated by Mr. Francis BaUy^
that, in the refined computations relative to the pendulum, the formulae for the
redaction to a vacuom are inaccurate, and that, in consequence, we do not yet
pneit^ know the length of a second pendulum. See Phil. Transac. 1832.
224 PENDULUMS. [part II.
When pendulums oscillate in equal times in different places,
we have
gig, :: I : l^.
For the variations of gravity in different latitudes, see formulie
(XVII., XVIII., XIX.) in the preceding section.
18. If the arcs are not indefinitely short, let v denote the versed
sine of the semiarc of vibration ; then
t = , y^i(l + i r + ^j 0 + &c) .... (VI.)
In which, when the seraiarc of vibration does not exceed 4 or 5
degrees, the third term of the series may be omitted.
If the time of an oscillation in an indefinitely small arc be 1 second,
the augmentation of the time will be
for a seraiarc of 30° 001675
of 15^* 000426
of 10° 000190
of 5° 000012
of 2J° 000003
So that for oscillations of 2^° on each side of the vertical, the
augmentation would not occasion more than 2^' difference in a day.
19. If D denote the degrees in the semiarc of au oscillating
pendulum, the time lost in each second by vibrating in a circle
instead of the cycloid, is ; and consequently the time lost in
a whole day of 24 hours, or 24 x 60 X 60 seconds, is ^ D^ nearly.
In like manner, the seconds lost per day by vibrating in the arc of
A degrees, is J A^. Therefore, if the pendulum keep true time in
one of these arcs, the seconds lost or gained per day, by vibrating in
the other, will be 4 (I>~ — A^). So, for example, if a pendulum
measure true time m an arc of 3 degrees, it will lose 11 1 seconds
a day by vibrating 4 degrees ; and 26 seconds a day by vibrating
5 degrees : and so on.
20. If a clock keep true time very nearly, the variation in the
length of the pendulum nccessaiy to correct the error will be equal
to twice the product of the length of the pendulum, and the error
in time divided by the time of observation in which that error is
accumulated.
If the pendulum be one that should beat seconds, and ^, the daily
variation be given in minutes, and n be the number of threads in an
inch of the screw which raises and depresses the bob of the pendulum,
then X = =■ — ^ i = '05434 n ^ = X « 'i » nearly, for the
24 X 60 ^^ ^
number of threads which the bob must be raised or lowered, to make
the pendulum vibrate truly.
21. For civil and military engineers, and other practical men, it is
CHAP. III.] CENTEB OF OSCILLATION. 225
highly useful to have Vk portable pendidum^ made of painted tape with
a brass bob at the end, so that the whole, except the bob, may be
rolled up within a box, which may be enclosed in a shagreen case.
The tape is marked 200, 190, 180, 170, 160, &c., 80, 75, 70, Qli,
OO, at points, which being assumed respectively as points of suspen
Qon, the pendulum will make 200, 190, &c., down to 60 vibrations
in a minute. Such a portable pendulum may be readily employed
in experiments relative to falling bodies, the velocity of sound, &c.
22. If the momentum of inertia (§ .5, page 221) of a pendu
lum, whether simple or compound, be divided by the product of
the pendulum's weight or mass into the distance of its center of
gravity from the point of suspension for axis of motion), the quotient
will express the distance of the center of oscillation from the same
point (or axis).
23. Whatever the number of separate masses or bodies which
constitute a pendulum, it may be considered as a single pendulum,
Dvhose center of gravity is at the distance d from the axis of suspen
sion, or of rotation : then, if K' denote the momentum of inertia of
tfmt body divided by its mass, the distance O from the axis of
rotation to the center of oscillation, or the length of an equivalent
mraple pendolnm, will be
= £±^.. (VII.)
a
24. To find the distanoe of the center of oscillation from the point
^^w axis of suspension, experimentally. Count the number, n, of
^^ecillations of the body in a very short arc in a minute ; then
O^l^ (VIII.)
Tims, if a body so oscillating made 50 vibrations in a minute;
^'^O^j^^SS'S^ inches.
Or, O = 89 j^ fy in inches, / being the time of one oscillation in a
^ry small arc.
If tlie arc be of finite appreciable magnitude, the time of oscilki
^n must be reduced in the ratio of 8 + versm of semiarc to 8,
'ore the rule is applied.
25. From the foregoing principles are derived the following ex
sessions for the distances of the centers of oscillation for the several
suspended by their vertices and vibrating flatwise, vis. :—
(1.^ Bi^t line or very thin cvlinder, O =  of its length.
(2.) IsMceles triangle, O 3=  of its altitude.
(3.) Circle, O ==  radius.
(4.) Common parabola, = 4^^^ altitude.
* For tome curiout and vahiabla theorems, by Professor ^try, for the re
^taecioo oi viknitSani ia the air to those in a VAoiuiin, tee Mr. F. Bailya peper
*>rerTed to in the preoeding note.
22i} CKNTKR OF USCILLATION. [PAKT II.
(5.) Any parabola, O = X its altitude.
3 m + 1
Bodies vibrating laterally or sideways, or in their own plane :
(6.) In a circle, O = } of diameter.
(7.) In a rectangle suspended by one angle, O =  of diagonal.
(8.) Parabola suspended by its vertex, O = 4 a^tis + J parameter.
(9.) Parabola suspended by middle of its base, O = ^ ^^^^ ~^ i
parameter.
,,^ N T r . 1 3arc X rad
(10.) In a sector of a circle, O = = — = — .
^ 4 chord
/,, X T ^  • (ra<l of base)*
(11.) In a cone, O == 4 axis 4 ^ ; — .
^ ^ 5 axis
2 rad^
(12.) In a sphere, O = rad f rf f = j: ; where d is the
5 (a + rad)
length of the thread by which it is suspended.
(13.) If the weight of the thread is to be taken into the account,
we have the following distance between the center of the ball and
that of oscillation, where B is the weight of the ball, d the distance
between the point of suspension and its center, r the radius of the
ball, w the weight of the thread or ^ire, and 1 the distance of the
center of oscillation from the center of gravity, we have
^^ aw^^B)4r''^}w(2dr±d^
(«jfB)</ — r» ^ "^
Or, if B be expressed in terms of w considered as a unit, then
^ = ^^ (X.)
(14.) If two weights W, W,, be fixed at the two extremities of a
rod of given length, S being the center of motion between W and
W, ; then, if d equal the distance of the weight W from S, D equal J
the distance of W, from S, and m the weight of a unit in length of ^
the rod, we shall have
^ ■" wD« + 2W,D — m</ — 2W</ ^ *^
the radii of the balls being supposed very small in comparison witbrJ
the length of the rod.
(15.) In the bob of a clock pendulum, supposing it two equaT^
spheric segments joined at their bases, if the radii of those bases b^^
each = ^, the height of each segment c, and d the distance from th^ j
point of suspension to the center of the bob, then is
'=w ''^'AV/" <^")
which shows the distance of the center of oscillation below the ccnteas'J
of the bob.
CHAP. III.] CENTER OF OSCILLATION. 227
If r the radius of the sphere he known, the latter expression
becomes
' "t:^^" <™''
(16.) Let the length of a rectangle be denoted by /, its breadth by
2 Wy the distance (along the middle of the rectangle) from one end to
the point of suspension by ^ then the distance O, from the point of
mspensioD to the center of oscillation, ^iill be
whether the 6gare be a mere geometrical rectangle, or a prismatic
metallic plate of uniform density. It follows from this theorem, that
a plate of 1 foot long and  of a foot broad, suspended at a fourth of
% foot from either end, would vibrate as a half second pendulum.
AIMS tliat a plate a foot long, ^^ of a foot wide, and suspended at
1^ of m foot from the middle, would vibrate 30,469 times in 5 hours.
Jknd henoBy ike length of a foot may he determined experimentcUly
hif vibratume.
(17.) If a thin rod» say of a foot in length, have
& ball of an inch diameter at each end, A and B,
&nd a moveable point of suspension, S; then the
tame of oteiUation of such a pendulum may be
wnade as long as we please ^ by bringing the point
of suspension nearer to the middle of the rod.
Or, if tbe point of suspension be fixed, the dis
tance O (and consequently the time of oscillations
'^'phicb is as >/0) may be varied by placing A
vi«arer to or farther from 8. And this is the prin
ciple of the Metronome^ by wliich musicians some
times regulate their time.
(18.) If the weight of the connecting rod be evanescent witli
^e^ard to the weight of the balls A and B ; then if R equal the radius
^f the larger ball, r that of the smaller, D and d the distances of their
■■'oipcctivc centers from 8 : we shall have
^__ B«(5D» + 2R^)hrM5(/''»H2r^)
^ 5(DR»).rfr') ^^^'^
When R and r are equal, this becomes
O = (D + rf) + ^ . ^^ (XVI.)
(19.) If the minor and major axes of an ellipse (or of an elliptical
P«e of wood or metal) be as 1 to >/ 3, or as 1000 to 1732 ; then,
Q 2
228 COMPENSATION TRNDULUMS. [PART IF.
if it be suspended at one extremity of the minor axis, the center of
oscillation will be at the other extremity of that axis, or its oscilla
tions will be performed in the same time as those of a simple pendu
lum whose length is eqnal to the minor axis.
The fnime ellipse also possesses this curious and useful, property^
Tiz. :— That any segment or any £one of the ellipse cut oflf by lines
parallel to the major axis, whether it be taken near the upper part
of the minor axis, near the middle, or near the bottom of the same,
will vibrate in the same time as the whole ellipse^ the point of sus
pension being at an extremity of the minor axis.
26. It is evident from § 1 7, page 228, that pendulums in differ
ent latitudes require to be of different lengths, in order that they
may perform their vibrations in the same time; but besides this
there is another irregularity in the motion of a pendulum in the same
place, arising from the different degrees of temperature. Heat ex
panding, and cold contracting the rod of the pendulum, certain
small variations must necessarily follow in the time of its vibration ;
to remedy which, Tarious methods have been invented for construct
ing what are commonly called compensation pendulums^ or atich as
shall always preserve the same distance between the center of oscil
lation and the point of suspension ; and of these we shall describe
two or three.
Comf^ensalion pendulums have received different denominations,
from their form and materials, as the gridiron pendulum^ mercurial "^
pendulum, &c.
27. The gridiron pendulum consists of five rods of steel, and four m ,i
of brass, placed in an alternate order, the middle rod being of steel, ..^ f J,
by which the pendulum ball is suspended ; these rods of brass andf^ d
steel are placed in an alternate order, and so connected with eachn^h
other at their ends, that while the expansion of the steel rods has s^ a
tendency to lengthen the pendulum, the expansion of the brass rods .Ezds
acting upwards tends to shorten it. And thus, when the lengths o ^n^oi
the brass and steel rods arc duly proportioned, theur expansions an<» mznd
contractions will exactly balance and correct each other, and so pre^^»e
serve the pendulum invariably of the same length. Sometimes 3, t w^~ 7,
or 9 foda, are employed in the construction of the gridiron penduKi^v .u
lum ; and zinc, silver, and other metals may be used instead of bras=%.^BS8
and steel.
28. The mercurial pendulum was invented by Mr. Graham, dt^w^ an
eminent clockmaker, about the year 1715. Its rod was made ^ of
brass, and branched towards its lower end, so as to embrace a cyliB^ci Ji
dric glass vessel 13 or 14 inches long, and about 2 indiea diamet er ^^r ;
which, being filled about 12 inches deep with mercury, forms iJ^B ie
weight or ball of the pendulum. The height of the mercury in ti^We
glass being so proportioned to the length of the rod that its expansi^*^^^
and contraction exactly balanced the expansion and contraction ^^1^
the pendulum rod, and preserved the distance of the center of osc ^■'Z'
lation from the point of suspension invariably the same.
This kind of pendulum fell entirely into disuse soon after Graham's
/
CHAP. III.] GYBATION AND ROTATION. 229
time ; but it has lately been reniclopted with considerable success by
practical astronomers. A very instractiTe paper on its principles,
construction, and use, has been published by Mr. F, Bailjfy in vol. i.
part 2, Memoirs of the Astronomical Society of London; in which
paper is also contained an extensive and valuable table of the expan
sion of different substances by beat.
29. Reid's compensation pendulum is a recent invention of Mr.
Adam Beid, of Woolwich, the construction of which is as follows : —
A N (fig. 189) is a rod of wire, and Z Z a hollow tnbe of zinc, which
slips ou the wire, being stopped from falling off by a nut N, on which
it rests ; and on the upper part of this cylinder of zinc rests the
heavy ball B: now the length of the tube ZZ being so adjusted to
the length of the rod A N, that the expansions of the two bodies shall
be equal with equal degrees of temperature; that is, by making the
length of the zinc tube to that of the wire, as the expansion of tlie wire
is to tliat of zinc, it is obvious that the ball B will in all cases pre
serve the same distance from A; for just so much as it would descend
by the expansion of the wire downwards, so much will it ascend by
the expansion of the zinc upwards, and consequently its vibrations
will in all temperatures be equal in equal times.
30. Drummond s compensation pendulum was proposed by an artist
of that name, in Lancashire, more than 70 years ago. A bar of the
same metal with the rod of the pendulum, and of the same thick
ness and length, is placed against the back part of the clock case ;
from the top of this a piece projects, to which the upper part of the
pendulum is connected by two fine pliable chains or silken strings,
which just below pass between two plates of brass whose lower
edges will alwavs terminate the lengtli of the pendulum at the upper
end. These plates are supported on a foot fixed to the back of the
case. This bar rests upon an immoveable base on the lower part of
the case, and is braced into a proper groove, which admits of no
motion any way but that of expansion and contraction in length by
heat and cold. In this construction, since the two bars are of equal
magnitude and of the same material, their expansions and con trac
tions will always be eqnal and in opposite directions ; so that one
will serve to correct and annihilate the effects of the other.
CENTER OP OYBATION AND THE PRINCIPLES OP ROTATION.
1. The center of gyration is that point in which, if all the matter
contained in a revolving system were collected, the same angular
velocity would be generated in the same time by a given force acting
at any plaee as would be generated by the same force acting similarly
ID tbe body or system itself.
When the axis of motion passes through the center of gravity,
tlien is the center called the principal center of gyration.
2. The distance of the center of gyration from the point of sus
pensioD or the axis of motion, is a mean proportional between the
distances of the centers of osdllation and gravity from the same point
or axis.
r"
230 GYRATION AND ROTATION. [PART 11.
If 6 equal the distance of tbe center of gravity from the point of
suspension, O the distance of tbe center of oscillation, and R the
distance of the center of g\Tation from the same point, then we have
R = >/go (XVII.)
3. The distance R of tlie center of gyration, from the center or
axis of motion, in some of the most useful cases, is given below.
In a circular wheel of uniform thickness ... R = rad >/ \,
111 the periphery of a circle revolving about) _^ A / X
the diameter j v s*
In the plane of a circle ditto R =  rad.
In the surface of a sphere ditto R =rad s/ §•
In a solid sphere ditto R = rad ^/t =^ i^rnearly.
In a plane ring formed of circles whose radii) /** "*"
are R, r, revolving about its center j ^\/ 2~
In a cone revolving about its vertex R = J ^ 3* ^' "+" J »''•
In a cone revolving about its axis R = rv^j^y.
In a paraboloid R = r ^^ i
/ R' + f^
In a straight lever whose arms are R and r, R = a / — r ; •
^ . V srR + r)
4. If the matter in any gyrating body were actually to be placed
as if in the center of gyration, it ought either to be disposed in the
circumference of a circle whose radius is R, or at two points R, R^
diametrically opposite, and each at the distance R from the center.
5. By means of the theory of the center of gyration, and the values
of R = ^, thence deduced, the phenomena of rotation on a fixed axis ^
become connected with those of accelerating forces : for then, if a ,^
weight or other moving ])ower P act at a radius r to give rotation to ^y
a body whose weight equals W, and the distance of whose center of "^
gyration from the axis of motion equals ^, we shall have for the ^^^
accelerating force, the expression
/=P^^ (^v"^>
and consequently for the space described by the actuating weight or "tk^di
power P, in a given time ty we shall have the usual formulse (V.) and f» jd
(VII.), page 214,
introducing the above value of/.
6. In the more complex cases, the distance of the center of gyra ^
tion from the axis of motion may best be computed from an experi ^"
ment. Let motion be given to the system, turning upon a horiatontal^^
axis, by a Mcight P acting by a cord over a pulley or wheel whose^^^
radius equals r, fixed upon the same axis, and let s be the space ^^^
through which the weight P descends in the time /, the proposed ^^
body whose weight is W turning upon the same axis with the wune— ^
angular velocity ; then
i
CHAP. III.] AXES OP ROTATION. 231
K=,.^€£f!lz^ ()
Example. A body which weighs 100 lbs. turns upon a horizontal
axis, motion being communicated to it by a weight of 10 lbs. hanging
from a very light wheel of 1 foot diameter. The weight descends
2 feet in 3 seconds. Required the distance of the center or circle of
gyration from the axis of motion.
" Here, potting y = 32, instead of 32^, we obtain as an approxima
tiye result,
R = ^ /32xl0x9xi4xl0xi ^ ^.33^3 ,^^^
V 4 X 100
I. When the impulse communicated to a body is in a line passing
through its center of gravity, all the points of the body move forward
with the same velocity, and in lines parallel to the direction of the
impulse communicated. But when the direction of that impulse docs
not pass through the center of gravity, the body acquires a rotation
OD an axis, and also a progressive motion, by which its center of
gravity is carried forward in the same straight line, and with the
same velocity, as if the direction of the impdse had passed through
the center of gravity.
The progressive and rotatory motion are independent of one
another, each being the same as if the other had no existence.
8. When a body revolves on an axis, and a force is impressed,
tending to make it revolve on some other, it will not revolve on
either, but on a line in the same plane with them, dividing the angle
which they contain, so that the sines of the parts are in the inverse
ratio of the angular velocities with which the body would have
revolved about the said axes separately.
9. A body may begin to revolve on any line as an axis that passes
through its center of gravity, but it will not continue to revolve per
manently about that axis, unless the opposite rotatory forces exactly
balance one another.
This admits of a simple experimental illustration. Suspend a thin
circular plate of wood or metal by a cord tied to its edge, from a hook
to which a rapid rotation can be given. The plate will at first turn
upon an axis which is in the continuation of the cord of rotation ; but
as the velocity augments, the plane will soon quit that axis, and
revolve permanently upon a vertical axis passing through its center of
gravity, itself having assumed a horizontal position.
The same will happen if a ring be suspended, and receive rotation
in like manner. And if a flexible chain of small links be united at
its two ends, tied to a cord and receive rotation, it will soon adjust
itself so as to form a ring, and spin round in a horizontal plane.
Also, if a flattened spheroid be suspended from any point, how
ever remote from its minor axis, and have a rapid rotation •:ivcn it, it
nill ultimately turn upon its shorter axis ])08itod vcrticully. This
evidently serves to confirm the motion of the earth upon its shorter
axis.
232 CENTHAL FORCES. [PABT II
10. In every body, however irregular, tbere are three axes of per
mancnt rotation, at right angles to one another. These are callei
the principal axes of rotation; and they have this remarkable pro
perty, that the raomentam of inertia with regard to any of them i
either a maximum or a minimum.
CENTRAL FORCES.
1 . Centripetal force is a force which tends constantly to solicit o
to impel a body towards a certain fixed point or center.
2. Centrifugal force is that by which it would recede from such
center, were it not prevented by the centripetal force.
3. These two forces are, jointly, called central forces.
4. If W denote the weight of a body moving in a circle* whoa
radius equals r, with the velocity t?, its centrifugal force scy will hm
f'"^ ■ ■ ■■ (•)
where g equals the force of gravity.
If t equal the time of one revolution, and ir = 31 41 59, then
/=i^=!^w («,.,
5. When a body describes a eirde by means of a force directed
its center, its actual velocity is every where equal to that which,
would acquire in falling by the same uniform force through half fc
radius.
6. This velocity is the same as that which a second body woi^
acquire by falling through half the radius, whilst the first described
portion of the circumfereuce equal to the whole radius.
7. In equal circles the forces are as the squares of the times »
versely.
8. If the times are equal, the velocities are as the radii, and C
forces are also as the radii.
9. In general, the. forces nre as the distances or radii of the drcf
directly, and the squares of the times inversely.
1 0. The squares of the times are as the distances directly, and tl
forces inversely.
11. Hence, if the forces are inversely as the squares of the di
tances, the squares of the times are as the cubes of the distance
That is,
if F :/ :: rf« : D«, then T : f ;: D' : d\ . . (XXII.)
12. The right line that joins a revolving body and its center of a
traction, called the radius vector^ always describes eqnal arcaa in eqa
times, and the velocity of the body is inversely as the perpendical
drawn from the center of attraction to the tangent of the t^nrve
the place of the revolving body.
13. If a body revolve in an elliptic orbit by a force directed to o
of the foci, the force is inversely as the square of the distance: «
CHAP. III.] CENTRAL FORCES. 233
the mean distances and the periodic times have the same relation as
in Art. 11. This eomprtkenda the case of the planetary motions.
14. If the force which retains a hody in a curve increase in tlie
simple ratio as the distance increases, the body will still describe an
ellipse; but the force will in this case be directed to the center of the
ellipse; and the body in each revolution will twice approach towards
it, and again twice recede from that point.
15. On the principles of central forces depends the operation of the
conical pendulum applied as a governor or regulator to steam engines,
water mills, &c.
This contrivance will be readily comprehended from fig. 1 90, where
A a is a vertical shaft capable of turning freely upon the sole a. C D,
e F, are two bars which move freely upon the center C, and carry at
their lower extremities two equal weights, P, Q; the bars CD, C F,
are united, by a proper articulation, to the bars O, H, which latter arc
attached to a nng 1, capable of sliding up and down the vertical
shaft A a. When this shaft and connected apparatus are made to
revolve, in virtue of the centrifugal force, the balls P Q fly out more
and more from A a, as the rotatory velocity increases: if, on the con
trary, the rotatory velocity slackens, the balls descend and approach
A a. The ring I ascends in the former case, descends in the latter:
and a lever connected with I may be made to reguhUe the energy of
the moving power. Thus, in the steam engine, the ring may be made
to act on the valve by which the steam is admitted into the cylinder;
to augment its opening when the motion is slackening, and recipro
cally diminish it when the motion is accelerated.
The construction is often so modified, that the fiying out of the
balls causes the ring I to be depressed,, and vice versd; but the gene
ral principle is the same.
Here, if the vertical distance of P or Q below C, be denoted by c/,
the time of one rotation of the regulator by ^, and 3 14 1593 by ^,
the theory of central forces gives
/ = 2 ^ /v/s^ = ^'^^^^^ >/ rf . . (XXIII.)
Hence, the periodic time varies as the square root of the altitude
of the conic pendulum, let the radius of the base be what it may.
Also, when ICQ = ICP=s 45°, the centrifugal force of each ball
is equal to its weight.
16. As the practical utility of the conical pendulum depends in a
great degree upon iu sensibility, or the change which must take place
m its Telocity before it will move the ring 1 to the required extent,
we subjoin the following formula for determining the weight of the
balls P and Q, for any degree of sensibility which may be required.
LetW equal the weight of both the balls, P the power required to
move the valve (or produce whatever effect may be required) when
applied to the ring in the direction of the spindle, J^, the number of
revolutions which the governor is intended to make per second, N^
the number of revolutions which the same must make to move the
234 iNQriitiEs cuNNfccTiiii WITH [part II.
ring I, </ as before the vertical distance from the plane of the balls
P and Q to the point C, b tlie distance CD or C F, / the distance
C P, a the distance of the point D from the axis of the spindle, c the
vertical distance from the plane joining the points D and F from the
point I, and r the radius of the circle described by the balls, or their
distance from the axis of the spindle ; then
'iJ^is? ^^^'^•>
If the distance C D is made equal to D I, so that the four rods form a
])aralleIogram, this last formula becomes
^^' = i..73r</?(N?3]^) • • • (^^^'^
17. In the foregoing formulte the center of oscillation of the balls
and system of rods is assumed to coincide with the center of the
balls, an assumption not strictly correct, although sufficiently so for all
practical purposes. Should, however, greater accuracy be required,
the true center of oscillation having been found, the following sub
stitutions must be made, when the preceding formulae will give an ac
curate result ; viz., for /, the distance from C to the center of oscilla
tion, and for r the distance of the center of oscillation from the ver
tical axis of the spindle.
INQUIRIES CONNECTED WITH ROTATION AND CENTRAL FORCES.
1. Suppose the diameter of a grindstone to be 44 inches, and iisa
weight half a ton ; suppose also that it makes 386 revolutions in tm
minute. What will be the centrifugal force, or its tendency to burst ^
44 386
Here the velocity = 75 X 31416 x ; = 74106;
then by substituting these values in formula (XX.) we have
^ 741062 X 5
/=  7  = 468 tons.
•^ 32 X tj
the measure of the required tendency.
2. If a fly wheel 12 feet diameter, and 3 tons in weight, revolivi
in 8 seconds : and another of the same weight revolves in 6 seconds
what must be the diameter of the last, when their centrifugal force
the same ?
By formula (XXI.) F : / : : ~ : ^. Therefore, since F is = ,,^
D d , Vt' 12 X 36 ^, ^
2^ = ;;}> ^»' « = ^P" = J54— = ^J ^^^^ ^^^ answer.
CHAP. III.] ROTATION AND CENTRAL FORCES. 235
3. If a fly of 12 feet diameter revolve in 8 seconds, and another
of the same diameter in 6 seconds : what is the ratio of their weights
when their central forces are equal ?
By § 7, page 232, the forces arc as the squares of the times in
versely when the weights are equal : therefore, when the weights are
unequal, tbey must be directly as the squares of the times, that the
central forces may be equal.
Hence » : W :: 36 : 64 :: 1 : 1^
That is, the weight of the more rapidly to that of the more slowly
revolving fly, must be as 1 to 1^, in the case proposed.
4. If a fly 2 tons weight and 1 G feet diameter, is suflicient to regu
late an engine when it revolves in 4 seconds; what must be the
weight of another fly of 12 feet diameter revolving in 2 seconds, so
that it may have the same power upon the engine ?
Hertf, by § 9, page 232, we must have — — = ^ ; therefore
40cwt. X 16 X 4 160 ,^, 1^ . ,
s   ^ ^ = ;« = 134 cwt., the weight
12 X 16 12 3 » 6
of tbe smaller fly.
NoUj'^K fly should always be made to move rapidly. If it be
intended for a mere Regulator, it should be near the fimt mover. If
it be intended to accumulate force in tbe workittg pointy it must not
be far separated from it.
5. Given the radius R of a wheel, and the radius r of its axle, the
weigbt of both, tp, and the distance of the center of gyration from the
axis of motion, ^ ; also a given power P acting at the circumference
of the wheel ; to find the weight W raised by a cord folding about
tbe axle, so that its momentum shall be a maximum. Here
^_ s/ (R^ P* f 2 R*Pg* w  h e^y*  f P»Rrg* f P*RV)— R'P— g*ig
Cor. 1. When R =s r, as in the case of the single fixed pulley,
(ben
W= >/(2P»R' + 2RPf*»+^»« h P»Re») — ^»— P.
R R"
Cor. 2. When the pulley is a cylinder of uniform matter e* = J R%
^•^d the expression becomes
W = VlR' (2 P« h 4 P » + 4 tP*)} — i w — P.
6. Let a given power P be applied to the circumference of a wheel,
^'liose radius equals R, to raise a weight W at its axle, whose radius
^^ r, it is required to find the ratio of R and r when W is raised with the
^^^eatest momentum ; the characters W and ^ denoting the same as in
^^^c last proposition.
236 COLLISION OF BODIES. [PART I
Here r = ; — r •
P (y r W)
Cor. Wlien the inertia of the machine is eTanesoent, with retp©
to that of P I W, then is r = R ^ / (\ + ^) — 1
7. In any machine whose motion accelerates, the weight ^ill I
moved with the greatest velocity when the velocity of the power is 1
that of the weight as 1 + P a / (^ + T^ ) ia to 1 ; the inertia i
the machine heing disregarded.
8. If in any machine whose motion accelerates, the deseent of oc
weight causes another to ascend, and the descending weight be give
the operation being supposed continually repeated, the effect vrill 1
greatest in a given time when the ascending weight is to the du
scending weight, as 1 to 1*618, in the case of equal heights; and
other cases when it is to the exact counterpoise in a ratio which^
always between I to 1 and 1 to 2.
9. The following general proposition with regard to rotatory nz
lion will be of use in the more recondite cases.
If a system of bodies be connected together and supported at «■
point which is not the center of gravity, and then left to descend
that part of their weight which is not supported, 2g multiplied \m
the sum of all the products of each body into the space it has p^
pendicularly descended, will be equal to the sum of all the prodiB
of each body into the square of its velocity.
Sect. IV. Percussion or Collision of Bodies in motion.
1. In the ordinary theory of percussion, or collision, bodies 
regarded as either hard^ sojly or elastic, A hard body is that wh—
parts do not yield to any stroke or percussion, but retains its fig*
unaltered. A soft body is that whose parts yield to any stroke
impression, without restoring themselves again, the shape of the be:
remaining altered. An elastic body is that whose parts yield to m
stroke, but presently restore themselves again, so that the body
gains the same figure as before the stroke. When bodies which he
been subjected to a stroke or pressure return only in part to i\m
original form, the elasticity is then imperfect: but if they rest*
themselves entirely to their primitive shape, and employ just as mu.
time in the restoration as was occupied m the compression, then
the elasticity perfect.
It has been customary to treat only of the collision of bodies p^
fectly hard or perfectly elastic : but as there do not exint in nat*
any bodies (which we know) of either the one or the other of th
kinds, the usual theories are but of little service in practical i*
CHAP. III.] COLLISION OP B0DTK8. 237
cbanics, except as tliey may suggest an extension to tbe actual cir
cumstances of nature and art.
2. Tbe general principle for determining tbe motions of bodies
from percussion, and wbicb belongs equally to both elastic and
nonelastic bodies, is this : viz. that there exists in the bodies the
same momentum, estimated in any one and the same direction, both
before the stroke and after it. And this principle is the immediate
result of the law of nature or motion, that reaction is equal to ac
tion, and in a contrary direction ; from whence it happens, that what
ever motion is communicated to one body by the action of another,
exactly the same motion does this latter lose in the same direction,
or exactly the same does the former communicate to the latter in tlie
contrary direction.
From this general principle too it results, that no alteration takes
place in the common center of gravity of bodies by their actions
upon one another ; but that the said common center of gravity per
severes in the same state, whether of rest or of uniform motion, both
before and after the impact.
3. If the impact of two perfectly hard bodies be direct, they will,
after impact, ekher remain at rest, or move on uniformly together
with different velocities, according to the circumstances under which
they met.
Let B and h represent two perfectly hard bodies, and let the velo
city of B be represented by V, and that of b by r, which may be
taken either positive or negative, according as h moves in the same
direction as B, or contrary to that direction, and it will be zero when
b is at rest. This notation being understood, all the circumstances
of the motions of the two bodies, after collision, will be expressed
by the formula :
, . BV±bv
velocity = —
B f
irliich being acoommodated to the three circumstances under which
9 may enter, become
„ V , , BV f 5t? / when both bodies moved in
(I.) velocity = p^^ \ the same direction
,,^ . , . BV — bv i when the bodies moved in
(II.) velocity = p^^ [ contmry directions
,^ ^ , . B 4 ft ( when the body b was at
(HI.) velocity =:^^ I rest.
Tbese formulsB arise fi^m the supposition of the bodies being per
fecOj hard, and consequently that the two after impact move on uni
formly together as one mass. In cases of perfectly elastic bodies,
other foraulflB have place which express the motion of each body
Mparately; as in the following proposition.
4. If the impact of two perfectly elastic bodies be direct, theii
relative velocities will be the same both before and aft^r impact, or
238 COLLISION OF BODIES. [pAHT
they will recede from eau^ other with the same velocity with whi
they met; that is, they will be equally distant, in equal times, bo^
before and after their collision, althou^ the absolute velocity of etu^^
may be changed. The circumstances attending this change of m.^
tion in the two bodies, using the above notation, are expr^eed in t~l^
two following formulae :
^~X — ^— = velocity of B . . . (IV.)
2BV + (B6)v , . ^,
^ ^— = velocity of 6 . . . (V.)
which needs no modification, when the motion of & is in the s^^
direction with that of B.
5. In the other case of b's motion, the general formulae become^
^^A!Ll_(Ezl*)^ = ,elocity«fB. (VI.)
2_LL^J!L:ii)^=velockyofft . (VII.)
when b moves in a contrary direction to that of B, which arises fr€>i
taking v negative. And
(B — b)y
^ — = velocity of B ( VIII.)
n n ^
   = velocity of ^ (IX.)
when b was at rest before impact, that is, when v = 0.
G. If a perfectly hard body B, (fig. 191,) impinge obliquely uf?^
a perfectly hard and immoveable plane A D, it will after coUi^'^
move along the plane in the direction C A.
And its velocity before impact
Is to its velocity after impact
As radius
Is to the cosine of the angle BCD.
But if the body be elastic it will rebound from the plane in t
direction C £, with the same velocity, and at the same angle wi 
which it met it, that is, the angle ACE will be equal to the an^
BCD.
7. The force with which a body impinging obliquely strikes
plane, is to the same if it had acted perpendicularly, as the sine ^
the angle (B C D) of incidence, is to radius.
8. In the case of direct impact, if B be the striking body, b tH
body struck, V and v their respective velocities before impact, V, ax^
«, their velocities afterwards; then the two following are genei^
formulce: viz.
CHAP. III.] ON THK MECHANICAL lOWFRS. 23.9
' v
r, = r f n
In these, if « = 1, they serve for nonelastic bodies ; if « = 2, for
bodies perfectly elastic. If the bodies be imperfectly elastic, n has
some intermediate Talue.
When the body struck* is at rest, the preceding equations become
V. = V  !Lll (XII.)
» B+b ^ '
r,=!^ (XIII.)
n = ^._(« +i) (XIV.)
BV ^ '
from which the value of n may be determined experimentally.
9. In the usual apparatus for experiments on Collision, balls of
different sizes and of various substances are hung from different
points of suspension on a horizontal bar. MAN (fig. 192) is an arc
of a circle whose center is S ; and its graduations, 1, 2,3, 4, 5, &c.,
indicate the lengths of chord lines measured from Uie lowest point D.
Any ball, therefore, as P, may be drawn from the vertical, and made
to strike another ball hanging at the lowest point, with any assigned
velocities, the height to which the ball struck ascends on the side
A M furnishing a measure of its velocity ; and from that the value of
n may be found from the last equation. Balls not required in an in
diridoal experiment, may be put behind the frame as shown at A
and B.
The cup C may be attached to a cord, and carry a ball of clay, &c.
when required.
Example, Suppose that a ball weighing 4 ounces strikes another
ball of the same substance weighing 3 ounces, with a velocity of ] 0,
^d communicates to it a velocity of 8 : what, in that case, will be
^he value of n ?
«eren= ' p^ = ^^^^ = ^j^ = 144375 the index of
the degree of elasticity; perfect elasticity being indicated by 2..
Sect. V. On the Mechanical Powers,
1. The most complicated machinery is nothing more than a com
bination, or constant repetition, of a few simple mechanical expe
dients for modifying and changing the direction of the several forces
240 MECHANICAL POWERS. [PART II.
or pressures, transmitted by them through the machine. A certain
pressure or force, being communicated to what is termed the first
or prime mover of the machinery, is thus transmitted through every
part of the machinery, being regulated and modified in such a man
ner as to produce the effect required, at what is termed the reork
ing point. Thus, by means of the machine termed a crane, a man, by
applying the pressure of his hand (amounting to perhaps about 30lb6.)
to the handle of the crane, is enabled, through the intervention of
the machinery, to raise an enormous weight, as, for instance, say
12,000 lbs., which, without some such expedient, would defy all his
efforts to move.
2. These elementary parts of which more complicated machinery
is composed, have been called the mechanical powers^ a term which
is liable to lead to a misapprehension of the effects which they are
really capable of producing. For, ^continuing the use of the previous
illustration,) although the man, through the instrumentality of the
crane, by exerting a ^rce of SOlbs., is enabled to lift 12,000lbs., or four
hundred times the amount, it will be found that his hand will move
four hundred times as fast as the weight, and, therefore, what advan
tage he may appear to eain in the weight which he is enabled to •«
lift, he really loses again by the length of time which it takes him to ^
raise it to any given height ; and which would be found by experi —
ment to be just what he would require to raise separately four hnn —
dred weights, each of dOlbs., to the same height.
3. The nnmber of the mechanical powers is usually reckoned to ^
be six : viz. the lever ^ the wheel and a^rfe, the pttUey, the indined^^
planey the wedge, and the screw.
4. In treating of these machines, we use the word power^ to denote ^s
the force which is supposed to be exerted at the origin of the machine, ^.i^
and the M'ord weight to denote the effect which that force produces^^
at the working point of the machhie : and we shall express them by "^
the letters P and \V respectively.
LEVERS.
1 . A lever is an inflexible bar, whether straight or bent, and sup^
posed capable of turning upon a fixed, unyielding point, called aful^
crum. There are three kinds of levers.
2. When i\\t fidcrum is between the power and the weight, as iia
fig. 193, the lever is said to be of \\\q first kind,
3. When the weight is between the power and the fulcrum, as in
fig. 194, the lever is of the second kind.
4. When the power is between the weight and the fulcrum, as in
fig. 1 95, the lever is of the third kind.
The bent lever, as employed in the operation of drawing a nail with
a hammer, is sometimes considered as a fourth kind, but is really a
lever of the first kind.
.5. In all these coses, where there is an equilibrium, it is indicated
by this general property, that the product of the weight into the dis
tance at which ft acts, is equal to the product of the power into the
CBAf.lu,] MECHANICAL POWERS. 241
distance at which it acts : tke distances being estimated in directione
P^'yndieidar to those in which the weight and power act respectively.
3T>», ID each of the three preceding figurcR,
P. AF = W. BF,
*• the power and weight are reciprocally as the distances at which
^cyact.
If, in fig. 193y for example, the arm A F were 4 times F B, 4 Ihs.
ftngfng at B would he halanced hy 1 Ih. at A ; and if A F were 5
Enes FB, 1 Ih. at A would halance 5 Ihs. at B; and so on.
6. If several weights hang upon a lever, some on one side of the
ilcnun, some on the other, then there will be an equilibrium, when
« sum of the products of the weights into their respective distances
1 one ade, is equal to the several products of weights and distances
la the other side.
Tor, the product of the weight into its distance from the fulcrum
\ is the same as its trioment about the point F, and therefore the
OTegmng proposition is a direct consequence of the principle of the
e«)Yiality of the moments of any forces in equilibrium, about any
fixed point, as explained at § 5, page 189.
7. When the weight of the lever is to be taken into the account,
F^oceed just as though it were a separate weight suspended at the
^We of its center of gravity.
8. When two, three, or more levers act one upon another in sue •
lesion, then the entire mechanical advantage which they afford, is
foond by taking, not the eum^ but the product of their separate ad
stages. Thus, if the arms of three levers, acting thus in con
P^on, are asStol, 4to1, and 5 to 1, then the joint advantage
» that of 3 X 4 X 5 to 1, or 60 to 1 : so that 1 lb. would, through
^ intervention, balance 60.
9. Id the first kind of lever the pressure upon the fulcrum = P
t W; in the other two it is = P '^ W.
10. Upon the foregoine principles depends the nature of scales
■Bd beams for weighing all bodies. For, if the distances be equal,
tko will the weights be equal also ; which gives the construction of
tlie common scales. And the Roman statera, or steelyard, is also a
Uifer^ but of unequal arms or distances, so contrived that one weight
anJy may serve to weigh a great many, by sliding it backwards and
Avwsrds to different distances upon the longer arm of the lever. In
tfce common halance^ or scales, if the weight of an article when
•soertained in one scale is not the same as its weight in the other,
Mr square root of the product of those two weights will give the true
metgit.
]]. From nnmeroos examples of the power and use of the lever,
we which shows its manner of application in the printingpresses of
dbe late Earl Stanhope may be advantageously introduced.
242
MECHANICAL POVfBBS.
[part II.
In the adjoining figure, let A BCD
be the general frame of the press,
connected bv the cross pieces N O,
DC. E is a center connected with
the frame by the bars E N, E R, E O.
To this center arc fixed a bar KL,
and a lever E F, to which the hand is
applied when the press is nsed.
Tiiere are also several other pieces
connected by joints at N, 6, 1, K, L,
M, O, H, which are so adjusted to
each other, that when the hand is applied to the lever EF at F, by ^\^^
f)ressing it downwards KL is brought into a horizontal line or paral ^JTg^
el to O U or D C, in which situation N I O, O M H, also form each omm j
straight line. It is evident that the nearer these different pieces, t^"i a^ 
above mentioned, are to a straight line the greater is the lever EF. ~=s^ f
in proportion to the perpendicular KS at the other end of the lever ^^ ^^
EK, formed by a perpendicular from K falling on F£ prodocedE:^ •?(/.
Consequently a small force applied at F will be sufficient to produce ^g
very great effect at K, when I K, K E are nearly in a straight line, an .^i^ajif
so on, for the other pieces above mentioned.
Hence the force applied by hand at F must be very considerable ^/e
in forcing down OH, which slides on iron cylindrical bars, or ? in
pressing any substance placed in the aperture PQ, between the be ^^
or plate and the frame DC.
This contrivance is now often introduced into mechanism, und er
the name of the toggle ^ or kneeJoint,
WHEEL AND AXLE.
1. The nature of this machine is suggested by its name. To it
may be referred all turning or wheelmachines containing wheels of
different radii ; as wellrollers and handles, cranes, capstans, \iir^ ^'
lasses, &c.
2. In the wheel and axle the mechanical property is the sam^ u
in the lever: viz. P . AC = W . BC (fig 196); and the reaaoim i»
evident, because the wheel and axle is only a kind of perpetrnv/
lever.
3. When a series of wheels and axles act upon each other, so ^
to transmit and accumulate a mechanical advantage, whether the
communication be by means of cords and belts, or of teeth nod
pinions, the weight will be to the power, not as the mm, but as tbe
continual product of the radii of the wheels to the continual product
of the radii of the axles. Thus, if the radii of the axles, a,ft,e,A<i
(fig. 197,) be each three inches, while the radii of tlie whedif
A, B, C, D, £, be 8, 6, 9, 10, and 12 inches respectively: tbtn
W:P::9xex9xlO x 12 : 3 X 3 X 3 X 3 X 3 :: 240:1. i
computation, however, in which the effect of friction is disr^arded.
PULLEY.
palley is a small wheel, commonly made of wood or brass,
of turning upon an iron axis passing through its center, and
a block, and the use of which is, that by means of a cord
ound its circumference, we are enabled to alter the direction
pce in any way that we choose. The pulley is either single
purpose, or combined with others to obtain a mechanical ad
. It is also either fixed or moveable, according as it is fixed
ilacc, or moves up and down with the weight or power*.
a power sustain a weight by means of a fixe<l pulley, as in
, the power and weight are equal.
lere are several different methods of combining pulleys to
or forming as they are termed separate systems of pulleys, for
g 4 mechanical advantage. ^
first system is shown in fig. 199, in which there are two
f pulleys, the upper fixed, and the lower attached to the
and rising with it ; only one continuous rope is employed,
lay be attached to either block, and passed successively round
dley. In this system, the weight sustained at W, is to the
sustaining it at P, as the number of ropes engaged between
blocks, is to 1 ; thus in fig. 199 a, the mechanical advantage,
I, and in fig. 199 5, it is 4 to 1.
be second system is when there are as many separate ropes
! are pulleys, each rope being attached to a fixed point at one
I, passing under one of the pulleys, has its other end attached
lock of the next pulley above it. This system is represented
•elul eombinaUon of the wheel and axle, a fixed and a moveable pulley
ted in the marginal dia
rhe loadf as of stones or >4b
» Imild a wall, is raited from F ^^
tkufs a rope BP Lis fixed UL
*_  I U I* I —
CHAP. III.] MECHANICAL POWERS. 245
Tkas, suppose the angle A B H was 30°, D B I 6*0% and con*
seqiiently ABD 90°: since tlic natural sines of 90% 60°, and 30°,
are 1, 866, and 5 respectively, or nearly as 100, 86*6, and 50; if
the heavy body weigh 100 lbs., the pressure upon AB would be
86'f> 1bs.i and upon BD 50 lbs.
This proposition is of very extensive utility, comprehending the
pressure of arches on their piers, of buttresses against walls, or upon
the ground, &c., because the circumstance of one of the pianos
becoming either horizontal, or vertical, will not affect the general
relation above exhibited.
WEDOB.
1. A wedge is a triangular prism, or a solid conceived
to be generated by the motion of a plane triangle parallel
to itself upon a straight line which passes through one of
its angular points. The wedge is called isosc^es^ rect
angtUar, or 9calene^ according as the generating triangle
IS isosceles, rightangled, or scalene. It is very fre
quently used in cleaving wood, as represented in the figure, and often
in raising great weights.
2. When a resistmg body is sustained against the face of a wedge,
by a force acting at right angles to its direction ; in the case of equi
Hbrium, the power is to the resistance as the sine of the semiangle
of the wedge, is to the sine of the angle which the direction of the
resistance makes with the face of the wedge; and the sustaining
force will be as the cosine of the latter angle.
3. When the resistance is made against the face of a wedge by a
body which is not sustained, but will adhere to the ])lace to which it
is applied without sliding, the power is to the resistance, in the ease
of equilibrium, as the cosine of the difference between the semiangle
of the wedge and the angle which the direction of the resistance
makes with the face of the wedge, is to radius.
4. When the resisting body is neither sustained nor adheres to the
point to which it is applied, but slides freely along the face of the
wedge, the jwwer is to the resistance as the product of the sines of
the semiangle of the wedge and the angle in which the resistance
ia inclined to its face is to the square of radius.
SCREW.
1. The screw is a spiral thread or groove cut round a cylinder, and
erery where making the same angle with the length of it. So that
if the surface of the cylinder, with this spiral thread on it, were un»
^Ided or developed into a plane, the spiral thread would form a
^rmight inclined plane, whose length would be to its height^ as the
^'rcnmference of the cylinder is to the distance between two threads
^f the screw : as is evident by considering that, in making one
'^und, the spiral rises along the cylinder the distance between the
^^o threads.
2, The energy of a power applied to turn a screw round, is to
^W force witli which it presses upward or downward (setting aside
246
MECHANICAL POWERS.
the friction), as the distance between two threads is to th
ference where the power is applied : viz., as the circumferei
is to the distance BI (fig. 205).
3. The endless screw^ or perpetual serew^ is one which
and tnms a toothed wheel DF (fig. 206), without a concave
screw; being so called because it may be turned for crei
coming to an end. From the diagram it is evident that
screw turns once round, the wheel only advances the d
one tooth.
4. If the power applied to the lever, or handle of a
screw, A B, be to the weight, in a ratio compounded of the
of the axis of the wheel, EH, to the periphery describe
power in turning the handle, and of the revolutions of the i
to the revolutions of the screw CB, the power will bn
weight. Hence,
5. As the motion of the wheel is very slow, a small p
raise a very great weight by means of an endless screw. A
fore the chief use of such a screw is, either where a great
to be raised through a little space, or where only a si
motion is wanted. For which reason it is very 8er\nceabh
and watches.
The screw is of admirable use in the mechanism of mi
and in the adjustments of astronomical and other instrun
refined construction.
6. The mechanical advantage of a compound machini
determined by analyzing its parts, finding the mechanical
of each part severally, and then blending or compound]:
ratios*. Thus, if m to 1, n to 1, r to 1, and e to 1,
separate advantages; then mnr s io 1, will measure the ad
the system.
* The marginal representation of a
common construction of a crane to raise
heavy loads, will serve to illustrate this.
By human energy at the handle a, the
pinion h is turned; that gives motion to
the wheel W, round whose axle, r, a cord
is coiled; which cord passes over the fixed
pulley, d^ and thence over the fixed triple
block, B, and the moveable triple block, P,
below which the load, L, hangs. Now,
if the radius of the handle be 6 times that
of the pinion, the radius of the wheel W
10 times that of iu axle, and a power
equivalent to 30 lbs. be exerted at a;
then, since a triple moveable pulley gives
a mechanical advantage of 6 to 1, we
shall have
30x6xl0x6» 10800 lbs.
and such would be the load, L. that might be raised by a powe
applied at a, were it not for the loss occasioned by friction.
HI.]
MBCHANICAL POWERS.
247
making such a calculation, the subjoined table, exhibiting at one
the ratio of the power to the weight, in all the simple mechani
wers, will be of service.
ipcioo oi Power*
and axle .
id wheels
§:—
tem (fig. 199)
lo. (fig. aoo)
to. (fig. 201)
9d plane
Ratio of
P : W.
■■^7
i.S.
r
n
1 :n
1 :^
1 : 21
Symbol
L equals the leverage
of the power, / that
of the weight.
I Resrad. of the wheel,
{ r that of the axle.
\ N •» No. of teeth in
I the wheel, n the No.
[ in the pinion,
) n equals the No. of
> pulleys, both fixed
) and moveable.
IX =» the length of the
plane, h its height.
Cx — the length of the
J side of the wedge, b
i the thickness of its
Chack.
^ e<a the drcumferenoe
V of the circle described
J by the power, and d
/the vertical disUnce
^between two threads.
ObMrvation.
/ In all the cases both
, the friction and weight
j of the machines them.
^ selves are neglected,
The strings are all
supposed to be pa
raUel.
When the power acts
in a direction parallel
to the plane.
f When the resistance
J inperpendiculartothe
y side of the wedge, the
(, wedge being single.
FABT II
CEAF. IT.
5«rT. L Gttfffrm DrfimiTm m M .
Er3^!»T^r:'3 cniL7nHe» nie Sacszae of sLe pumme and the
~ rivsL ic itin^iassx f iba&. a» v^iicr, aerrair, &c^ mnd that of
2. A ffwfKt jtmmi s a Vocj vintfe parts are rerr miDOte, vieldiag
to acT f!:«Te lAzrewvc XT«aK h . Xiov«icr saail), uid bj ao jielding
srre Mifff^mr t ira l ideas of a flaid bodr, bj com j
: h %o a beap of a^ ; b«t die xBiTkwiabuitj of ^^^^ flaiditv b? ii^p^i
aaj \jzA of 3Kc£u3caJ cosauaiitaoii. vill apfwar br eonsidenii^ twci ■ wi
9i tibe cRsaMcaaen ttpcrwrr to coofdtate a fluid bodj : I . That^i^=t
tbe parts. iMKvidataDdin* aar compmaoii. maj be mored in
tioa to eadk oiLer, aiih tbe smallest coooeiTabie foree, or will gire
DO mnuMt retu^mmct to mocioo wiibio the mass in anj directioD.
2. That the parts shall sravitate to each ocber, wfaerebr there is
eoostant teodencT to amnse tbemselTes about a common center, i
form a j^pKerical bodv ; which, as the parts do not resist motion, i^^^
easilj effected in soiali bodies. Henee the appearance of drops^^
alurajs takes p!ace when a jlmiJ is in proper circamstances. It i^^^
obrions that a body of sand can bj no means conform to thiMJiii^
drciraistances.
Differeot fluids hare different degrees of flaiditr, according to th^^
^KiIitT with which the particles maj be mored amongst each other. —
Water and mercary are classed anumg the most perfect floids— ^
Many floids hare a rcrr sensible degree of tenacity, and are therefor^^'
called viscous or imperfect fluids.
3. Fluids may be diTided into compressible and ineompressMe^ otf^
elastic and nonelastic fluids. A compressible or elastic fluid is on^
whose apparent magnitude is diminished as the pressure upon it vm
increaseil, and increued by a diminution of pressure. Such is air.^
and the different Tapours. An incompressible or nondastie flui<3
(called also a liquid) is one whose dimenuons are not sensibl/
affected by any augmentation of pressure. Water, mercury, oil, &c ^
are generally ranged under this class.
It has been of late years proposed to limit the application of the
term fluidi to those which are ekutiCy and to apply the word liquid
to such as are nonelastic.
CHAP. IV.] PKKSSURB OP FLUIDS. 249
4. The specific gravity or density of any solid or flaid body, is the
absolute weight of a known volume of that substance ; namely, of that
volume which we take for unity in measuring the capacities of bodies.
Sbct. II. Pressure and Equilibrium of Nonelastic Fluids.
1. Fluids press equally in all directions^ upwards, downwards,
aslant, or laterally.
This constitutes one essential difference between fluids and solids,
solids pressing only downwards, or in the direction of gravity.
2. The upper surface of a gravitating fluid at rest is horizontal.
3. The pressure of a fluid on every particle of the vessel containing
it, or of any oUier surface, real or imaginary, in contact with it, is
equal to the weight of a column of the fluid, whose base is equal to
tliat particle, and whose height is equal to its depth below the upper
BDifaoe of the fluid.
4. If, therefore, any portion of the upper part of a fluid be
Replaced by a part of the vessel, the pressure against this from below
niil be the same which before supported the weight of the fluid
i^emoTed, and every part remaining in equilibrium, the pressure on
the bottom will be the same as it would be if the vessel were a prism
Or a cylinder.
5. Hence, the smallest given quantity of a fluid may be made to
Produce a pressure capable of sustaining any proposed weight, either
oy diminishing the diameter of the column and increasing its height,
Or hy increasing the surface which supports the weight.
6 The perpendicular pressure of a fluid on any surface, whether
>rertica], oblique, or horizontal, is equal to the weight of a column of
^e fluid whose base is equal to the surface pressed, and height equal
^o the distance of the center of gravity of that surface below the
Opper horizontal surface of the fluid.
7. Fluids of different specific gravities that do not mix, will
^soanterbalance each other in a bent tube, when their heights above
tihe surface of junction are inversely as their specific gravities.
A portion of fluid will be quiescent in a bent tube, when the upper
surface in both branches of the tube is in the same horizontal plane,
or is equidistant from the earth's center. And water poured down
one branch of such a tube (whether it be of uniform bore throughout,
or not) will Ti»e to its own level in the other branch.
Thus, water may be conveyed by pipes from a spring on the side
of a hill, to a reservoir of equal height on another hill.
8. The ascent of a body in a fluid of greater specific grarity than
itself, arises from the pressure of the fluid upwards against the under
surface of the body
9. The center of pressure is that point of a surface against which
any fluid prestes, to which if a force equal to the whole pressure
250 PRESSURE OF FLUIDS. 
were applied it would keep the surface at rest, or balance
ency to turn or move in any direction.
10. If a plane surface which is pressed by a fluid be pn
the horizontal surface of it, and their common intersection be
as the axis of suspension, the centers of percussion and of
will be at the same distance /rom the axis,
11. The center of pressure of a parallelosram, whose upp
in the plane of the horizontal level of the liquid, is at § oj
(measuring downwards) that joins the middles of the two }
sides of the parallelogram.
12. If the base of a triangular plane coincides with tl
surface of the water, then the center of pressure is at the i
the line drawn /rem the middle of the base to the vertt
triangle. But, if the vertex of the triangle be in the uppe
of the water, while its base is horizontal, Ae center of pt
at ^ of the line drawn from the vertex to bisect the base.
Id. If in any closed vessel containing a fluid suppose
without weight, an opening or orifice be made and any pn
applied, that pressure will be equally distributed over tl
interior surface of the vessel; and if the fluid has a wei[
own, the pressure upon any point will equal the sum of tl
buted pressure and the pressure occasioned by the weight of
at that point.
ILLUSTRATIONS AND APPLICATIONS.
t. If several glass tubes of different shapes and
sizes be put into a larger glass vessel containing
water, the tubes being all open at top ; then the
water will be seen to rise to the same height in
each of them, as is marked by the upper surface
a c, of the liquid in the larger vessel.
2. If three vessels of equal bases, one cylindrical, the sec
siderably larger at top than at bottom, the third considerab
top than at bottom, and with the sides of the two latt<
regularly or irregularly sloped, have their bottoms moveable,
close by the action of a weight upon a lever ; then it will I
that when the same weight acts at the same distance upon t
water must be poured in to the same height in each vessel I
pressure will force open the bottom.
3. Let a glass tube open at both ends (whether cylindrio
does not signify) have a piece of bladder tied loosely over <
so as to be capable of hanging below that end, or of rising u
it, when pressed from the outside. Pour into this tube son
tinged red, so as to stand at the depth of seven or eight inci
CHAP. IV.] bramah's PBESS. 251
then imroerae the tuhe with its coloured water vertically into a larger
glass vessel nearly foil of colourless water, the hladder heing down^
wards, serving as a fiexihle bottom to the tube. Then, it will be
observed that when the depth of the water in the tube exceeds that
in the larger vessel, the bladder will be forced behw the tube, by the
excess of the interior over the exterior pressure : but when the ex
terior water is deeper than the interior, the bladder will be thrust up
within the tube, by the excess of exterior pressure : and when the water
in the tube and that in the larger vessel have their upper surfaces in
the same horizontal plane, then the bladder will adjust itself into a
fiat position, just at the bottom of the tube. The success of this
experiment does not depend upon the actual depth of the water in
the tube, but upon the relation between the depths of that and the
exterior water; and proves that in all cases the deeper water has
the greater pressure at its bottom, tending equally upward or
downward.
4. The hydrostaticcU paradaof^ as it is usually denominated, results
Prom the principle that any quantity of a nonelastic fluid, however
ma]}, may be made to balance another quantity; or any weight, as
arge as we please (§ 5, page 249). It may be illustrated by a
naohine, the hydrostatic heUow9^ which consists
»f two thick boards DC, F£, each about 16
r 18 inches diameter, more or less, covered
r connected firmly with leather round the
dges, to open and shut like a common bel
yy/v9y bot without valves; only a pipe A B,
boat 8 feet high, is fixed into the bellows
boTO F. Now, let water be poured into
lo pipe at A, and it will run into the bellows,
radoally separating the boards by raising the
pper one. Then, if several weights, as three hundredweights, be
lid upon the npper board, and water be poured in at the pipe till it
» fnJl, it will sustain all the weights, though the water in the pipe
bonld sot weigh a quarter of a pound : for the pipe or tube may be
a sniall as we please, provided it be but long enough, the whole
flTect depending upon the height, and not at all on the width of the
ipc, for the proportion is always this: —
As the area of the orifice of the pipe
is to the area of the bellows board,
so is the weight of water in the pipe, above D C,
to the weight it will sustain on the board.
5. In lieu of the bellows part of the apparatus, the leather of which
rould be incapable of resisting any very considerable pressure, the
ale Mr. Joseph Bramah used a very strong metal cylinder, in which
I piston moved in a perfectly air and water tight manner, by passing
.hroogh leather collars, and as a substitute for the high column of
ivaler be adopted a very small forcing pump, to which auy power can
be applied ; M>d thus the pressing column becomes indefinitely long.
252 biumah's press. [part ii.
although the whole apparatus is very compact and takes hut little
room. Figure 207 is a section of one of tliese presses, in which t is
the piston of the large cylinder, formed of a solid piece of metal
turned truly cylindrical, and carrying the lower hoard v of the press
upon it : u is the piston of the small forcing pump, being also a
cylinder of solid metal moved up and down hy the handle or lever w.
The whole lower part of the press is sometimes made to stand in a
case XX, containing more than sufficient water, as at y, to Bll both
the cylinders; and the suction pipe of the forcing pump u dip])ing
into this water will be constantly supplied. Whenever, therefore,
the handle w is moved upwards, the water will rise through the
conical metal valve jt, opening upwards into the bottom of the pump
u ; and when the handle is depressed that water will be forced
through another similar valve a, opening in an opposite direction in
the pipe of communication between the pump and the great cylinder
ft, which will now receive the water, by which the piston rod t will
be elevated at each stroke of the pump «. Another small conical
valve c is applied by means of a screw to an orifice in the lower part
of the large cylinder, the use of which is to release the pressure
whenever it may be necessary ; for, on opening this valve, any w^at ei^
which was previously contained in the large cylinder ft, will run off
into the reservoir y by the passage d^ and the* piston t will descend ^^
so that the same water may be used over and over again. Th(
power of such a machine is enormously great; for, supposing th
hand to be applied at the end of the handle tr, with a force of onl
10 pounds, and that this handle or lever be so constructed as t
multiply that force but 5 times, then the force nith which the pistoi
u descends will bo equal to 50 pounds: let us next suppose that th<
magnitude of the piston t is such, that the area of its horizontc
section shall contain a similar area of the smaller piston u 50 times
then 50 multiplied by 50 gives 2500 pounds, for the force wit
which the piston i and the lower board v of the press will rise. J
man can, however, exert ten times this force for a short time, am
could therefore raise 25,000 pounds ; and would do more if a greai
disproportion existed between the two pistons / and u^ and the lev
w were made more favourable to the exertion of his strength.
This machine not only acts as a press, but is capable of niai
other useful applications, such as a jack for raising heavy loads,
even buildings ; to the purpose of drawing up trees by their rooi
or the piles used in bridgebuilding.
To find the thickness of the metal in Bramah's press, to resi
certain pressures, Mr. Barlow gives this theorem, t = — — whe J
p =s pressure in lbs. per square inch, r = radius of the cylinder, i ==
its thickness, and c = 1 8000 Ihs. the cohesive power of a squ^^y
inch of cast iron.
£a;. Suppose it were required to determine the thickness of m^tsT
in two presses, each of 6 inches radius, in one of which the pressure
I
CHAP. IV.] FRBSSURS AGAINST SLUICEGATES, ETC. 253
may extend to 4278 pounds, in the other to 855G pounds per square
inch.
Here in the first case,
4278 X 6 .««,., t . ,
' = ,»^^^ 7x=^ = 1*87 inches, thickness.
18000 — 4278
In the second,
8556 X 6 . .o . L 11
^ = .^^^^ TTzzz^ = 5*43 inches, thickness.
18000 — 8556 '
The usual rules, explained helow (Art. 10), would make the latter
thickness douhle the former : extensive experiments are necessary to
tell whicli method deserves the preference.
6. If b equals the breadth, and d the depth of a rectangular gate,
or other surface exposed to the pressure of water from top to hottom ;
then the entire pressure is equal to the weight of a prism of water
inrhose content is ^ hd^. Or, if ^ and d be in feet, then the whole
pressure = 31 J ft d\ in lbs., or nearly = y\ b d\ in cwts.
7. If the gate be in the form of a trapezoid, widest at top, then,
i f B and b be the breadths at the top and bottom respectively, and d
the depth, the
whole pressure in lbs. = 31 J {^^ (B — b) + b} d^
whole pressure in cwts. = ^ {^^ (B — b) ^ b}d^ nearly.
8. The weight of a cubic foot of rain or river water, is nearly equal
^o ,\ cwt.
The pressure on a souare inch, at the depth of THiRty feet is very
nearly THiR/^it pounds.
The pressure on a square foot is nearly a ton at the depth of thirty
six feet. [The true depth is 3584 feet.]
The weight of an ale gallon of rain water is nearly 10 lbs., that
of an imperial ^Woii 10 lbs.
The weight of a cubic foot of •eawater is nearly ^ of a cwt.
These are all useful approximations ; the actual weight of a cubic
foot of distilled water is 6*25 lbs.
Thus, the pressure of rain water upon a square inch at the depth
of 3000 feet, is 1300 lbs.
And tlie pressure upon a square foot at the depth of 108 feet is
nearly three tons.
9. In the construction of dykes or embankments, both faces or slopes
should be planes, and the exterior and interior slopes should make
an angle of not less than OO"*. For, if A D' (fig. 208) be the exterior
slope, and the angle D^ A B be acute, E D' perpendicular to A B is the
direction of the pressure upon it; and the portion D'A E will pro
bably be torn off. But when DA is the exterior face, making with
ABan obtuse angle, the direction of the pressure falls within the
base, and therefore augments its stability.
10. The strength of a circular bason confining water requires the
consideration of other principles.
254 FLOATING BODIES. [PART II.
The perpendicular pressure against the wall depends merely on the
altitude of the fluid, without being affected by the volume. But, as
Professor Leslie remarks, the longitudinal effort of the thrust, or its
tendency to open the joints of the masonry, is measured by the radios
of the circle. To resist that action in very wide basons, the range
or course of stones along the inside of the wall must be proportion
ally thicker. On the other hand, if any opposing surface present
some convexity to the pressure of water, the resulting longitudinal
strain will be exerted in closing the joints and consolidating the
building. Such reversed incurvation is, therefore, often adopted in
the construction of dams.
Upon similar principles, the thickness of pipe$ to convey water,
must vary in proportion to — , where h is the height of the head of
water, d the diameter of the pipe, and c the measure of the cohe«on
of a bar of the same material as the pipe, and an inch square.
A pipe of cast iron^ 15 inches diameter, and \ of an inch thick,
will be strong enough for a head of 600 feet.
A pipe of (Hik of tlie same diameter, and 2 inches thick, woold
sustain a head of 180 feet.
Where the cohesion is the same, t varies B&hd; or as HD : T :: :
hd : t^m the comparison of two cases *.
Example, What, then, must be the respective thicknesses of pipes
of cast iron and oak, each 10 inches diameter, to carry water from sh
head of 360 feet ?
Here, 1st, for cast iron:
HD (= 600 X 15) : T (= ^) :: A rf (= 860 x 10)
860 X 10 X S 10800 s r • ,_
eOOT"! 5'ir4 = 36000 = ^^ ^^ *^ '°^*^
2ndly, for oo^:
HD (a= 180 X 16) : T ( = 2) :: A J (= 360 x 10)
860 X 10 X ft ^, „ «« . ,
i8o^ri5 = ^ = *=^^^°^^^
Sect. III. Floating Bodies.
1. If any body float on a fluid, it displaces a quantity of the fluid
equal to itself in weight.
* To ascertain whether or not a pipe is strong enough to sustain a proposed
pressure, it is a good custom amongst practical men to empk>y a «^l^e«lo#,
usually of an indi in diameter, and load it with the proposed weight, and s
surpluM determined by practice. Then, if the proposed pressure be aj^ied
interiorly, by a forcing pump, or in any other way, if the pipe remain sound in
all its parts after the safetyvalve has yielded, such pipe is regarded as sufficiently
strong.
The aetual pressures upon a pipe of any proposed diameter and head, may
evidently be determined by a similar method.
CHAP. IV.J
PARKY S FLOODOATE.
255
'2. Also, the centers of gravity of the body and of the fluid di8
pkieed, must, when the body is at rest, be in the same yertical line.
8. If aressel contun two fluids that will not mix (as water and
mercurj), and a solid of some inteimediate specific gravity be im
mened under the surface of the lighter fluid and float on the heavier;
the part of the solid immersed in the heavier fluid, is to the whole
solid aB the difference between the spedflc gravities of the solid and
the lighter fluid, is to the difference between the specific gravities of
the two fluids.
4. The buoyancy of casks, or the load which they will carry with
out sinking, may be estimated by reckoning 10 lbs. avoirdupois to the
ale gallon, or 8 lbs. to the wine gallon.
5. Tlie buoyancy of pontoons may be estimated at about kalf a
kwiredwei^ for each cubic foot.
Thns a pontoon which contained 06 cubic feet, would sustain a
load of 48 cwt. before it would sink. This is an approximation, in
which the difference between ^ and , that is, ^^ of the whole
weight, is allowed for that of the pontoon itself.
6. The principles of buoyancy are very ingeniously applied in
Mr. Farcy's tdf acting floodgate. In the case of common sluices to
a niildam, when a sudden flood occurs, unless the miller gets up in
the nip;ht to open the gate or gates, the neighbouring lands may be*
come inundated ; and, on the contrary, unless he be present to shut
th«n np when the flood subsides, the milldam may be emptied and
the water lost which he would need the next day. To prevent either
of these occurrences, Mr. John Farey, whose talent and ingenuity are
well known, has proposed a ielfacting floodgate^ the following de
scription of which has been given in the Mechanics Weekly Journal.
— j
_. .
=^^IS
w^
^^
r^^
_B
gj:^::x
Fi
,..,,..,^„,.„„.,,...jp.., — —
wLt^
m~r
V  
m <i
ter
'^
t^
.^ ,^i^^V^ — ^ — ^^^—  — ^ " —
1
SA / "
IV :/ 1 ,_>^J:^^
1
4 \y ! \ V^i^p^ —
r
p
[:■ '*~^
256 SPECIFIC GRAVITIES. [PART II.
A A represents a vertical section of a gate poised upon a horizon
tal axis passing rather above the center of pressure of the gate, so as
to give it a tendency to shut close: a a is a lever, fi.xed perpendicu
lar to the gate, and connected by an iron rod with a cask, 5, which is
floated whenever the surface of the water rises to the line B, D, which
is assumed as the level of the wear, or milldam, B,C,£, F, in which
the floodgate is placed : by this arrangement it will be seen that when
the water riscK above the dam, it floats the cask, opens the gate, and
allows the water to escape until its surface subsides to the proper
level at B, D; the cask now acts by its weight, when unsupported
by the water, to close the gate and prevent leakage. The gate should
be fitted into a frame of timber, H,K, which is set in the masonry of
the dam. The upper beam H, of the frame being just level with the
crown of the dam, so that the water runs over the top of the gate at the
same time that it passes through it: to prevent the current disturbing
the cask, it is connected by a small rod, e, at each end, to the upper
beam, H, of the frame, and jointed in such a manner as to admit of
motion in a vertical direction.
7> By means of the same principle of buoyancy it is, that a hollow
ball of copper attached to a metallic lever of about a foot long, is
made to rise with the liquid in a watertub, and thus to close the
cock and stop the supply from the pipe, just before the time when
the water would otherwise run over the top of the vessel.
8. This property, again, has been successfully employed in pulling
up old ])i]e8 in a river where the tide ebbs and flows. A barge of
considerable dimensions is brought over a pile as the water begins to
rise: a strong chain which has been previously fixed to the pile by a
ring, &c. is made to gird the barge and is tlien fastened. As the tide
rises the vessel rises too, and by means of its buoyant force draws up
the pile with it.
In an actual case, abarge 50 feet long, IS feet wide, 6 deep, and draw
ing *2 feet of water, was employed. Here, 50 x 12 x (6 — a) x ?
= ^Q X ^^ X 16 ^ ^^^ X 7 = 1344 + 274 = 1371f cwt.= 66J
tons nearly, the measure of the force with which the barge acted
upon the pile.
Sect. IV. Specific Gravities.
1. If a body float on a fluid, the part immersed is to the whole
body, as the specific gravity of the body to the specific gravity of the
fluid.
Hence, if the body be a square or a triangular prism, and it be
laid upon the fluid, the ratio of that portion of one end which is im
mersed, to the whole surface of that end, will serve to determine the
specific gravity of the body.
2. If the same body float upon two fluids in succession, the parts
immersed will be inversely as the specific gravities of those fluids.
CRAP. ly.] SPECIFIC GRAVITIES. 257
3. The weight which a body loses when wholly immersed in a
fluid is equal to the weight of an equal bulk of the fluid.
When we say that a body loses part of its weight in a fluid, we do
not mean that its abtoluie weight is less than it was before, but that it
is partly supported by the reaction of the fluid under it, so that it
requires a less power to sustain or to balance it.
4. A body immersed in a fluid ascends or descends with a force
equal to the difference between its own weight and the weight of an
eqoa] bnlk of fluid ; the resistance or viscidity of the fluid not being
considered.
5. To Jind ike specific aravity of a fluids or of a solid. — On one arm
of a balance suspend a globe of lead by a fine thread, and to the other
fasten an equal weight, which may just balance it in the open air.
Immerse the globe into the fluid, and observe what weight balances
it then, and consequently what weight is lost, which is proportional
to the specific gravity as above. And thus the proportion of the
Q)ecific gravity of one fluid to another is determined by immersing
the globe successively in all the fluids, and observing the weights lost
in each, which will be the proportions of the specific gravities of the
^mds sought.
This same operation determines also the specific gravity of the
Solid immerged, whether it be a globe or of any other shape or bulk,
Supposing that of the fluid known. For the specific gravity of the
lluia is to that of the solid, as the weight lost is to the whole
^eight.
Hence also may be found the specific gravity of a body that is
lighter than the fluid, as follows :
O. To find the specific gravity of a solid that is lighter than the fluid y
^i^stPOieTj in which it MjEm/.— Annex to the lighter body another that
i^ much heavier than the fluid, so that the compound mass may sink
in tbe fluid. Weigh the heavier body and the compound mass sepa
««tel J, both in water and out of it ; then find how much each loses
ivi urater, by subtracting its weight in water from its weight in air;
^nd subtract the less of these remainders from the greater.
Then, As this last remainder,
: the weight of the light body in air,
: : the specific gravity of the fluid,
: the specific gravity of that body.
7. The specific gravities of bodies of equal weight are reciprocally
proportionar to the quantities of weight lost in tbe same fluid. And
tience is found the ratio of the specific gravities of solids, by weighing
in the same fluids masses of them that weigh equally in air, and
noting the weights lost by each.
8. Instead of a hydrostatic balance^ a hydrostatic steelyard is now
frequently employed. It is contrived to balance exactly by making
the shorter end wider, and with an enlargement at the extremity.
The shorter arm is undivided, but the longer arm is divided into
short equal divisions: thus, if that longer arm be 8 inches long, it
258 CAPILLARY ATTRACTION. [PART II.
may be divided into 400 parts, the divisions wmmeneing at A (fig. 209V
Then, in using this instrument, any convenient weight is suspended
by a hook from a notch at the end of the scale A. The body who$e
specific gravity is to be determined, is suspended from the other am^^
by a horsehair, and moved to and fro till an equilibrium is produced^ _ ,
Then, without altering its situation at D on the beam, it is immenev:^
in water, and balanced a second time by sliding the couDterpois^^^
from A, say to C.
Here, evidently, the weight in water : weight in air : : B C : B Am^ ^
and the loss of weight in water : weight in air :: AG : AB.
^ , weight in air AB .^
Consequently, — — = —  = specific gravity.
Joss A C
With such an instrument nicely balanced upon a convenient ped^^^
tal, I find that the specific gravities of solids are ascertainable b^^n^^
with greater facility and correctness than with any hydrostaH^t^^
balance which I have seen *.
A copious table of the specific gravities of various substances "99111
be found in the Appendix, Table X.
Sect. V. On Capillary Attraction,
1 . If two plates of glass be set up vertically in a shallow vessel,
containing a coloured liquid, in such a way as to touch along the
edges AB, fig, 210^ and forming a very acute anele vidth each other,
the coloured liquid will rise between the two plates to a height in
versely as the distance between them ; from which it follows, that its
surface will form a rectangular hyperbola, whose two asymptotes are
the edge of the plate A B and the surface of the li(^uid. And if a
glass tube, the bore of which is small, be immersed in the liquid, it
will rise within the tube to a greater height than the surfiEMse of the
liquid without the same.
2. The cause of this phenomenon is an attractive force existing
between the liquid and the glass, and which is termcfd capillary at
traction, from capiUuSy the Latin for hair, because it only Utkes place
in tubes, the bores of which are small, resembling hairs.
8. The heights to which fluids rise in capillary tubes are inversely
as their diameters : and they are found to nse to the same height in
a vacuum, as in the air.
4. In a tube r^n^^ ^^ ^^ ^^^^ ^° diameter, water rises 5*3 inches;
and as the height is inversely as the diameter, their product will be a
constant quantity, therefore 5*3 X *01 = *053 = ^, which may be
taken to represent the attractive force between water and glaaa. The
following are the values of ^, according to different experimenters :— 
* We owe this coDtrivance to Dr. CotUtiy of Philaddpfaia.
CHAP. IV.] CAPILLARY ATTBACTION. 259
Sir I. Newton 020
Sir D. Brevvster 033
M.M.Gay Lussac 046
Mr. Atwood '053
5. The height at which the fluid is supported hy capillary attrac
tion in a tube of Yar3dng diameter, is that due to its size at the sur
face of the liquid, without any regard to the dimensions or shape of
the lower portion. Thus, in a vessel of the form shown in figure 211,
terminating in a capillary tube, the liquid will be supported at the
same heifi;ht in this tube (and also fill the whole of the space abc of
the TesscI) as if the tube had been of uniform dimensions throughout
its whole length, and every where equal to its upper portion. The
water, however, contained in the lower portion of the vessel a & c, is
in this case supported by the pressure of the atmosphere, as it ceases
to be supported in a vacuum.
6. Different liquids rise to different heights in capillary tubes of
the aaxne bore, depending upon the attraction between them and
^lass. The following are Dr. Brewster's results for several fluids,
irith a tube '0561 inches in diameter : —
Height of Value of the
aaoent in inchef. constant q.
Water 587 0327
Water, very hot 537 0301
Muriatic acid 442 0248
Nitric acid 395 0222
Spermaceti oil 392 0220
Olive oil 387 '0215
Oil of turpentine 333 0187
Alcohol 317 0178
iEther 285 0160
Sulphuric acid 200 0112
7. The internal diameter of a uniform capillary, or other small
tube, may be found in the following manner. Let the tube be weighed
when empty, and again when filled with mercury, and let w be the
difference of those weights in troy grains, / the length of the tube in
inches; and d its diameter, then d = '019252 \/j
Thos, if the difference of the weights were 500 grains, and the
length of the tube were 20 inches: we should have d = *010252
y^^ = 019252 X 5 = 09626 of an inch*
* Tlie same thing may easily i>e aeoomplithed thot t — Let a cone of box wood,
0r of bran, be very aoeunitely turned, or about 6 inches in length, and the dia
naecer of its base aboat a quarter of an inch ; and let its curve surface i>e very
aecoratelj marked with a series of parallel rings, about a twentieth of an inch
aa i mde r, from its vertex to its base. Insert this cone carefully in the cylinder
(ao that their axes shall coincide) as in fig. 212 : then it will be as V A i Va : t
A B : eft; where, as the ratio of V A to V a is known by means of the equi
distant rings on the iorfaea, and A B is known, a b becomes determined.
82
260 SPFLUBNC£ OF FLUIDS. [pABT II.
CHAP. V.
HYDRODYNAMICS.
HydrodynamicB is that part of mechanical science which relates to
the motion of nonelastic fluids, and the forces with which they act
upon bodies.
This branch of mechanics is the most diflicult, and the least ad
vanced : whatever we know of it is almost entirely due to the re
searches of the modems.
Could we know with certainty the mass, the figure, and the num
ber of particles of a fluid in motion, the laws of its motion might be
determined by the resolution of this problem, viz. to find the motion
of a proposed system of small free bodies acting one upon the other
in obedience to some given exterior force. We are, however, ▼cry
far from being in possession of the data requisite for the solution of
this problem. We shall, therefore, simply present a few of the most
usually received theoretical deductions ; and then proceed to state
those rules which have resulted from a judicious application of theory
to experiment.
Skct. I. Motion and Effluence of Liquids.
1 . A jet of water, issuing from an orifice of a proper fonn, and
directed upwards, rises, under favourable circumstances, nearly to the
height of the head of water in the reservoir; and since the particles
of such a stream are but little influenced by the neighbouring ones,
they may be considered as independent bodies, moving initially with
the velocity which would be acquired in falling from the height of the
reservoir. And the velocity of the jet will be the same whatever
may be its direction.
2. Hence, if a jet issue horizontally from any part of the side of
a vessel standing on a horizontal plane, and a circle be described
having the whole height of the fluid for its diameter, the fluid wiU
reach the plane at a distance from the vessel, equal to that chord of
the circle m which the jet initially moves.
Thus, if AS (fig, 213) be the upper surface of the fluid in the ves
sel, B the place of the orifice, CF the horizontal plane on which the
fluid spouts, then CF is equal to £D, the horizontal chord of the
BPFLUXNCE OF FLUIDS. 261
le diameter is A C, passing throagh B. It is therefore evi
tbe orifice from which the fluid will spout to the greatest
I situate at G, half the height of the fluid, and also, that if
ires be made at equal distances ahoTC and below 6, the jet
om both will strike the plane C £ in the same point.
m a cylindrical or prismatic vessel empties itself by a small
3 Telocity at the surface is uniformly retarded; and in the
nptying itself, twice the quantity would be discharged if it
I full by a new supply.
the quantity dischsrged is by no means equal to what would
lole orifice, with this velocity. If the aperture is made sim
lin plate, the lateral motion of the particles towards it tends
i the direct motion, and to contract the stream which has
ifice, nearly in the ratio of two to three. So that, in order
le quantity discharged, the section of the orifice must be
to be diminished from 1 00 to 62 for a simple aperture, to
ipe of which the length is twice the diameter, and in other
»rding to circumstances.
m. a syphon, or bent tube, is filled with a fluid, and its
mersed in the fluids of different vessels, if both surfaces of
are in the same level, the whole remains at rest ; but if
the longer column of fluid in the syphon preponderates,
■easure of the atmosphere forces up the fluid from the higher
til the equilibrium is restored ; and the motion is the more
be difference of the levels is greater: provided that the
^ht of the tube above the upper surface be not more than
loise to the pressure of the atmosphere.
le lower vessel be allowed to empty itself, the syphon will
mining as long as it is supplied from the upper, and the
it descends the further below the vessel. In the same
e discharge of a pipe, descending from the side or bottom
i vessel, would be increased almost without limit by length
notch or sluice in form of a rectangle be cut in the ver
of a vessel full of water, or any other fluid, the quantity
loagb it will be ^ of the quantity which would flow through
nifice, placed horizontally at the whole depth, in the same
reasel being kept constantly full.
ivr«Deiit in the oonstniction of the syphon has been lately proposed
pnv Mechanics^ Magazine^ and by M. burUem at Paris. It might
mtageouftly used if constructed on a larse scale, for lowering the
B4laiBS or canals. The improvement in toe present syphon is, that
pipe is enlarged to the same diameter as that of the syphon, and
lened out to something of a funnel shape, as in fig. 214. In
J action, the short arm is immersed in the water as in the usual
bottom of the long arm is closed, the exhausting pipe is then filled
hjr the funnel^aped mouth. On the bottom of the long arm being
Mtter flows out, and exhausts the air from the syphon, when the
rh wished to be emptied flows out in a continual stream.
z&
i
262 PIPB8 AND CANALS. [PART II.
8. If a short pipe elevated in any direction from an aperture in t
condnit, throw the water in a parabolic curre to the distance or range
R, on a board, or other horizontal plane passing through the orifice^
and the greatest height of the spouting fluid above that plane, be H,
then the height of the head of water above that conduit pipe, may be
found nearfy: vis. by taking 1st, 2 cot £ = ^; and 2ndly, the alti
2 n
tude of the head A = i R x cosec 2 E.
Example. Suppose that R = 40 feet, and H = 18 feet. Then
— == ~ = Mllllll =2cot60*'57': and A = i R x cosec2E
= 20 X cosec 121* 5V = 20 x 1177896 = 2355792 feet, height
required.
Note. This result of theory will usually be found about ^ of that
which is furnished by experiment
Sbci. II. Motion of Water in Conduit Pipes and Open Canals^
over Weira^ 4^.
1. When the water from a reservoir is conveyed in long horizontal
pipes of the same aperture, the discharges made in equal times are
nearly in the inverse ratio of the square roots of the lengths.
It is supposed that the lengths of the pipes to which this rule is
applied, are not very unequal. It is an approximation not deduced
from principle, but derived immediately from experiment. [Bossut,
tom. ] 1, $ 647, 648. At § 673, he has given a table of the actual
discharges of waterpipes, as far as the length of 2340 toises, or
14,950 feet English.]
2. Water running in open canals, or in rivers, is accelerated iu
consequence of its depth and of the declivity on which it runs, till
the resistance, increasing with the velocity, becomes equal to the
acceleration, when the motion of the stream becomes uniform.
It is evident that the amount of the resisting forces can hardly be
determined by principles already known, and therefore nothing
remains but to ascertain, by experiment, the velocity corresponding
to different declivities, and different depths of water, and to try, by
multiplying and extending these experiments, to find out the law
which is common to them all.
The Chevalier Du Buat has been successful in this research, and
has given a formula for computing the velocity of running water,
whether in close pipes, open canals, or rivers, which, though it may
be called empirum^ is extremely useful in practice. Principea
d'Hydrauligue. Professor Robison has given an abridged account of
this book, m his excellent article on Rivers and Waterworics, in the
JEnejfdopcadia Britannica.
Let V be the velocity of the stream, measured by the feet it moves
over in a second ; R a constant quantity, vis., the quotient obtained
CHAP, v.] PIPES AND CANALS, 263
by dividiog the area of the transverse section of the stream, expressed
in square feet, by the boundary or perimeter of that section, minus
the saperficial breadth of the stream expressed in linear feet.
The mean velocity is that with which, if all the particles were to
move, the discharge would be the same with the actual discharge.
The line R is <^Ied by Dn Buat, the raditi$; and by Dr. Robison,
the k^raulie mecm depth. As its affinity to the radius of a circle
eeenifl greater than to the depth of a river, we shall call it, witli the
former, the radiui qfihe section,
LasUy, let 8 be the denominator of a fraction which expresses the
slope, the numerator being unity, that is, let it be the quotient ob
tained by dividing the length of the stream, supposing it extended in
^ straight line, by the difference of level of its two extremities : or,
virhich is nearly the same, let it be the cotangent of the inclination
or slope.
3. The above denominations being understood, and the section, as
well as the velocity, being supposed uniform, we have
When R and 8 are very great,
^ = »'{sn^.4' ""''^^ ^"^
The logarithms understood here are the hyperbolic, and are found
by multiplying the common logarithms by 2*3025851 ; or more easily
by the method described at page 94.
The slope remaining the same, the velocities are as n/r — y^.
The velocities of two great rivers that have the same declivity, are
as the square roots of the radii of their sections.
If R is so small, that n/R — ^j^ = 0, or R = ^j, the velocity
will be nothing; which is agreeable to experience; for in a cylindric
tube R =s ^ Uie radius ; the radius, therefore, equals twotenths ;
so that the tube is nearly capillary, and the fluid will not flow
through it.
The vdodty may also become nothing by the declivity becoming
so small, that
if  is less than — — —, or than Vn^h of an inch to an English
8 600000' ;^ ^
mile, the water will have sensible motion.
4. Ib a river, the greatest velocity is at the surface, and in the
middle of the stream, from which it diminishes towards the bottom
2S4 PIPES AND CANALS. [PABT II.
and the sides, where it is least. It has been found by experiments
that if from the square root of the Telocity in the middle of the
stream, expressed in inches per second, unity be subtracted, the
square of the remainder is the velocity at the bottom.
Hence, if the former velocity be = 9, the velocity at the bottom
c=r— 2v^t?4l (HI.)
5. The mean velocity, or that with which, were the whole stream
to move, the discharge would be the same with the real discharge, is
equal to half the sum of the greatest and least velocities, as computed
in the last proposition.
The mean velocity is, therefore, =»— ^/cf^ (IV.)
This is also proved by the experiments of Dn Buat.
6. Suppose that a river having a rectangular bed, is increased by
the junction of another river equal to itself, the declivity remaining
the same ; required the increase of the depth and velocity.
Let the breadth of the river = 5, the depth before the junction </,
and after it d?; and in like manner, v and v^ the mean velocities
before and after; then ^ , is the radius before, and
b + 2d ' 6 + 2x
the radius after, so o = r — , supposing the breadth of the river
to be such, that we may reject the small quantity subtracted from R,
SOT R *
in formulce (I. and II.); and, in like manner, v^ = r— !.
Then, substituting for R and R^, we have
^^30T , f~hd
S* ^ ^ cH 2<^'
307 / vof
and c. =T X A/ — .
Multiplying these into the areas of the sections 5 </ and ior, we
have the discharges, viz.,
307 hds/hd ^^ 307 hxs/hx
Now the last of these is double of the former ; therefore,
hx^hx 2hds/hd x" 4rf»
>/^ f 2rf' 'fth2« 642rf'
and
t TTk ^ = 1 ri J a cubic equation which can always
be resolved by Cardan's rule, or by the approximating method given
at page 82.
As an example, let ft = 10 feet, and </ = 1, then x^ — J « = ^*>,
and X = 1*4882, which is the depth of the increased river. Hence
we have 1488 x t>j = 2 r, and 1'488 : 2 : : «? : Cj, or t? : ©j : : 37
to 50 nearly.
CHAP, v.]
CANALS, RIVKBS, &C.
265
When the water in a river receives a permanent increase, the
depth and the velocity, as in the example above, are the first things
that are augmented. The increase of the velocity increases the
action on the sides and bottom, in consequence of which the width is
aaemented, and sometimes also, but more rarely, the depth. The
velocity is thus diminished, till the tenacity of the soil or the hard
ness of the rock affords a sufficient resistance to the force of the
^water. The bed of the river then changes only by insensible
degrees, and, in the ordinary language of hydraulics, is said to be
permanent, though in strictness this epithet is not applicable to the
course of any river.
7' When the sections of a river vary, the quantity of water
semaining the same, the mean velocities are inversely as the areas of
^he sections. This must happen, in order to preserve the same
quantity of discharge. (Pla^air's Outlines,)
8. The following table, abridged from Du Buat, serves at once to
compare the surface, bottom, and mean velocities in rivers, according
t:o the formul® (III. and IV.)
VdocUies of Rivers.
VRI/K31TY IN INCHES.
VELOCITY IN INCHES.
Sm&ce.
Bottom.
Mean.
Snr&ce.
Bottom.
Mean.
4
1
25
56
42016
49008
8
3342
567
60
45509
52754
la
6071
9036
64
49
565
16
9
125
68
52505
60252
20
12065
16027
72
56025
64012
24
15194
19597
76
59568
67784
28
18421
23210
80
63107
71553
32
21678
26839
84
66651
75325
36
25
305
88
70224
79112
40
28345
34172
92
73788
82894
44
31742
37871
96
77370
86685
48
35151
41570
100
81
905
52
38564
45282
9. The Icnowledge of the velocity at the bottom is of the greatest
use for enabling us to judge of the action of the stream on its bed.
Every kind of soil has a certain velocity consistent with the sta
bility of the channel. A greater velocity would enable the waters to
tear it up, and a smaller velocity would permit the deposition of more
movable materials from above. It is not enough, then, for the sta
bility of a river, that the accelerating forces are so adjusted to the
260 CANALS, RIVBBSy &C. [PART II.
size and figure of its channel that the current may be in train : it most
also be in equilibrio with the tenacity of the channel.
We learn from the obseryations of Du Buat, and others, that a
velocity of three inches per second at the bottom will jast begin to
work upon the fine clay fit for pottery, and howerer firm and com
pact it may be, it will tear it up. Yet no beds are more stable
than clay when the velocities do not exceed this: for the water
soon takes away the impalpable particles of the superficial clay,
leaving the particles of sand sticking by their lower half in the
rest of the clay, which they now protect, making a very perma
nent bottom, if the stream does not bring down gravel or coarse
sand, which will rub off this very thin crust, and allow another layer
to be worn off; a velocity of six inches will lift fine sand; eight
inches will lift sand as coarse as linseed ; twelve inches will sweep
along fine gravel ; twentyfour inches will roll along rounded pebbles
an inch diameter; and it requires three feet per second at the bottom
to sweep along shivery angular stones of the size of an egg. (i2o6i
son on Rivers,)
1 0. In the elbow or bend of a river, the velocity is always greater
near the concave than the convex side.
1 1 . The swell occasioned by the piers of a bndge, or the sides of a
cleaning sluice which contract the passage by a given quantity, for a
short length only of the channel, may be determined when the ver
tical section of the river and the velocity of the stream are known, in
the following manner. Let v be the velocity of the stream, inde
pendently of the effect of the bridge, r the section of the river, and a
the amount of the sections between the piers; let 2^, instead of
being taken at 64j, be reckoned 58'6, to accord with the results of
experimental contractions through arches of bridges, &c., and let $
be the slope of the bed of the river, or the sine of its angle with the
horizon ; then Du Btiat (tom. i. p. 225) gives for the swell or rise
(R) of the stream in feet, which will be occasioned by the ob*
struction.
(3^0) (GI); •• ■<^'
12. The value of s will, of course, be different in different cases;
but if we assume ^ or '05, as a mean value, it will enable us to
compute and tabulate results, which, though they cannot be pre
sented as perfectly correct, may be regarded as exhibiting a medium
between those that will usually occur; and will serve to anticipate
the consequences of floods of certain velocities, when constrained to
pass through bridges which more or less contract the stream.
CHAP, v.]
SWELLS 0C0A8I0NED BY BRIDGES.
267
TaUe of the Rise of Water occaeumed hy Piere of Bridges^ or
other Contractions^.
Amoum of otHtTiictiaafi f^inupved with th* vortical HetJon of the Hiver.
MOth.
a^iothi.
3lOlhA
40)>thi. aioth».
Bltrthj.
7'imhi.
aieth..
o^iotiii.
4
Pruporliotml Riie of W»tcr, in f«rt and decimals.
feel.
tbe^
rwt
fwt.
f«t.
fc«t.
fMt.
feeu
fMt.
<K11S7
(M)377
^HMlMr
0*[IS2
OS<H!t
03631
(1^78
l^gw
mam
*os77
<]^)IW5
flisai
OS 102
«':t'^#4«
ivn^^ 1
1199S
ia37a
\i1\m
\OTtUll«T7
J flex>di.
^NHT?
IM*4
0^1 in
DaiSlfl
«^JfP7
IHlfiHT
MSiM>
4i»MI
201 A>4
(H*70l
i>18»
frasTf
mi^
{*^iti
7iM«i
3't7&5
rTjan
3tm^>
ftUfiS
(^'^709
i^sim
fl'STUS
i*4aafl
26<ifpfi
5i>if>2
il'ftlff
4ft 1555
> Violent
1 flfxirti.
criAsa
0^19
v\m
iijyi?5
:!Mifiti
d7i^
iMtlH
tts7sie
tPi«7«
C^4<m:
(H»iSl
i57a>
sri579
46511
aSL^iTe
nmm
Ji7'TiW>
riiii78
Oeiia
11BB4
iteso
3*25*
aawT
H^AIM
^Mm
li3tM±i
I t'lJiit'iiPilly
033iO
^"'N>54
IMi^JO
255Ge
A'W^
7'MT3
14^4777
34a!J4e
UlJSAi
J. vjolcnt
10
0^41 19
\mn
imj^
^'li^ie
s^asw
9^tJt»
J77M1
4a*i+MJ
iTk^um
1 floodi. ,
13. It will be eyident, from an inspection of this table, that even
in the case of ordinary floods, old bridges with piers and starlings,
occupying 6 or 7tentbs of the section of the river, will produce a
swell of 2, 3, or more feet, often overflowing the river^s banks and
occasioning moch mischief. Also, that in violent floods, an obstmc
tion amounting to Ttenths of the channel, will cause a rise of 7 or 8
feet, probably choking up the arches and occasioning the destruction
of the bridge. Greater velocities and greater contractions produce a
rapid augmentation of danger and mbchief ; as the table obviously
shows.
14. The same principles and tabulated results «erve to estimate
the fall from the higher to the lower side of a bridge, on account of
an ebbing tide, &c. Thus, for old London Bridge, where the
breadth of the Thames is 926 feet, and the sum of the water ways
at low water only 236 feet, the amount of the obstructions was 690
feet, about 7 tenths of the entire section : so that a velocity of
3 1 feet per second would give a fall of nearly 4 feet, agreeing with
the actual result.
At Rochester Bridge, before the opening of the middle arches, the
piers and starlings presented an obstruction of 7tenths, and at the
time of greatest fall> the velocity 100 yards above bridge exceeded
6 feet per second. This, from the table, would occasion a fall of
more than 67 feet ; and the recorded results vary from 6 to
7 feet
At Westminster Bridge, where the obstructiona are about lsixth
of the whole channel, when the velocity is 2^ feet, the fall but
little exceeds half an inch: a result which the table would lead
us to expect.
•. A timilar table was computed by Mr. Wright of Durham, more than
ftfty jmn ago, and inierted in the first edition of Dr. Hutton*B treatiM on
Bndget ; but it it not constructed upon a correct theory.
268 eytblwbin's bbsults. [part u.
15. Mr. Eytelwein^ a German mathematiciany has devoted much
time to inquiries in hydrod3mamic8. In his investigations he has
paid attention to the mutual cohesion of the liquid moleculee, their
adherence to the sides of the vessel in which the water moves, and
to the contraction experienced by the liquid vein when it issues from
the vessel under certain circumstances. He obtains formulse of the
utmost generality, and then applies them to the motion of water;
1st, in a cylindric tube; 2ndly, in an open canal.
16. Let d be the diameter of the cylindric tube EF (fig. 215),
h the total heieht F G of the head of water in the reservoir above the
orifice F, and ? the length £ F of the tube, all in inches : and v equal
the velocity in inches per second with which the fluid will issue
from the orifice F ; then
V •
^'iV'f^ <"•>
which, multiplied into the area of the orifice, will give the quantity
discharged per second.
1 7. Let d = the diameter of the pipe in inches, Q = the quantity
of water in cubic feet discharged through the pipe per minute, / =
the length of the pipe in feet, and h = the difference of level between
the surface of the water in the reservoir and at the end of the pipe,
or the head; then, any three of these quantities being given, the
fourth may be determined from the follo\idng formulae: —
^= ^'O^^^QC^^^^^^ (VII.)
« == V^tt^^^^^t:^ (VIII.)
hd'
•0448 (/ + 42 d)
I = ^^' ,  4 2rf (IX.)
•0448 Q'* ^ ^
. 0448 Q2(/ + 42 rf) .„^
^= ^^5 (^^
These formulae are more convenient when expressed logarithmic
ally, and then become
log (/ = J {2logQ +26515 + log(/ + 42(/) logA} (XL)
logQ = I {logh + 5\ogd^ 26515  log(/ + 42flO} ... (XIL)
log / =log;i+5logrf2'65152logQ,[neglecting — 42flri(XIII.)
log A = 2 logQ+ ■26515 + log (/ + 42 (/)  5 log rf (XIV.)
18. When a pipe is bent in one or more places, then if the squares
of the sines of the several changes of direction be added into one
sum «, the velocity v will, according to Lang^dorf^ be foond by
the theorem
CHAP.
v.]
bttblwbin's bbsults.
269
= a/^
SMdk
(XV.)
■^l.+ ids
/, k^ d^ and 9, being all in inches.
19. For open canals. — Let v be the mean velocity of the current
in feet, a area of the vertical section of the stream, p perimeter of
the section, or snm of the bottom and two sides, / length of the bed
of the canal corresponding to the fall hy all in feet : then
* = /^ 9582 — + 00111  0109 (XVI.)
The experiments of M. Bidone^ of Turin, on the motion of water
in canals, agree within the 80tli part of the results of computations
from the preceding formulae.
20. For apertures in the sides or bottom of vessels,— \^ q equals the
€]iiantity of water discharged in cubic feet per minute, v the velocity
of the effluent water in feet per second, through the aperture, a the
sreA of the aperture in square inches, and h the height from its center
to the surface of the water ; we have
v^Cs/h (XVII.)
y=4ie7ac ^h (XVIII.)
In which c is a constant quantity depending upon the nature of the
aperture, and the value of which for several different forms is con
tained in the following table.
Nature of the Orifices employed.
Ratio between
the theoretical
and real
Coefficients
for finding
the velocities
in Eng. feet.
For the whole velocity due to the height
For an orifice of the form of the contracted >
vein.
For wide openingi whose bottom is on a 1
kvd with that of the reservoir ; for sluices (
with walls in a line with the orifice ; for T
bridges with pointed piers.
For narrow openings whose bottom is on a^
levd with that of the reservoir; for
mailer openings in a sluice with side 
waDa ; for abrupt projections and square
piers of bridges. J
For openinp in sluices without side walls
For sui orinoe in a thin nlate
1 to 100
1 to 0969
1 to 0961
1 to 0861
1 to 0635
1 to 0621
1 to 0510
1 to 0671
1 to 0808
80458
7 8
77
69
51
50
41
54
65
When there is a short cylindrical pipe pro )
jeeting on the inside of the vessel, length (
9 to 4 times the bore, and rim like con f
tracted vein, not fulL J
Idem, when it runs fuU bore of water
Idem, when it does not project inside the \
VCMCL S
MOTION IN PIPI89 BTC.
[PA&l
r
t
I
Bore
of the
pipe in
inches.
'*"*'p^ :I?:?2'« oSTSToo « ^ ?.o»i^»«©2
.SO
1
S 2 2 ZS ^ * *^* *® "* '* •* ** ** ** ** '^ '^ ^ '^ '^
6 « © w lb r»b « ©• Oft » 00 r»^ « '^ ^
1
© © ep * «o © 00 ^© t^oo "* 00 cc* « »^r*9
t^^oot^««'<«eoeoc«G«e4p^i«^4^©©©c
a.
1
a
1
1"
© rt ©© *©©*(» rt ^ 5 (N :• »^ »« r <?•« «
00 — ©to©6>o©ooeo^oo©<«^^©o»aor<»cs
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fa
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fa
t>.N©eO©iO'*0 — C0M^»O» — — '*00«9»«
•^©o© ^©©r^ib&«©©©Qor«©>e<«'^ei»f
t^^eowe^N^ — » — — ©o©©oo©©©
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*: ^ lb 91 ^ © 00 r«»^ •« kid ^ CO eo 99 e« o« N ..
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»^— '*t>.©«eitcp»'?''r'»^T*'?'9f
;ii,:^QO©ib'ii"^ooeoN©i«'^'^*'^^<
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10 © <N 00 10 CO  © © r^o « '^ "^ « o'
;,«©,^^;^^©©©©©©©©<
©©©©©©oo©©o©©©e©
Bore of
the pipe
'in
inches.
'*"*'^ :ir;r2'N srs?co So ^ ^5 « t
CHAP, v.]
TABLBS, &C., FOB WEIRS.
271
Look for the Telocity of water in the pipe in the upper rew, and
io the colamn below it, and opposite to the given diameter of the
pipe studing in the last column, will be found the perpendicular
ha^ of a column or head, in feet, inches, and tenths, requisite to
orercome the friction of such pipe for 100 feet in length, and obtain
the giren Telocity.
22, TaUe containing the qttantity of Water discharged over an inch
vertical section of a Weir.
Depth of the
upper edge
of the waste
boaid below
theiur&ce
in English
inches.
Cnbic feet of
water discharged
in a minute by an
inch of the waste
board, according
to Du Buat's
formuls.
Cubic feet oi
water discharged
in a minute by an
inch of the waste
board, according
to experiments
made in Scotland.
Gallons
of
282 inches,
corresponding
with
results in
col. 3.
1
0403
0428
2621
2
1UO
1211
7417
3
2095
2226
13634
4
8226
3427
20990
5
4507
4789
20332
6
5925
6295
38357
7
7466
7983
48589
8
9122
9692
59364
9
10884
11664
70826
10
12748
13535
83164
11
14707
15632
95746
12
16758
17805
109055
13
18895
20076
122965
14
21117
22437
137427
15
23419
24883
152408
16
25800
27413
167906
17
28258
80024
183897
18
30786
82710
200350
^^ To the aboTe table, originally due to Du Buaty is added a third
^^QiDD, containing the quantities of water discharged, as inferred
J^Jffl experiments made in Scotland, and examined by Dr. Robison,
Jo found that they in general gaTe a discharge ^ greater than that
^lich » deduced from Du BucSs formulas. We would recommend
^ therefore to the engineer to employ the third column in his prac
^^ or the fourth if he wish for the result in gallons.
. Ifdiej be odd quarters of an inch, look in the tabic for as many
^^<ie« as the depth contains quarters, and take the eighth part of
^inswer. Thus, for 3 inches, take the eighth part of 24883,
^luch corresponds to 15 inches. This is 3*110.
23. The quantity discharged increases more rapidly than the width :
^ obUin a correct measure of it, if / be the width or length of tho
272 DI8CHAROX8 OVBB WEIBS. [PABT 11.
wasteboard in inches, take (i^^l) times the quantity for one inch of
wasteboard of the given depth, from the preceding table.
In the preceding table it is supposed that the water from which the
discharge is made is perfectly stagnant; but if it should happen to
reach the opening with any velocity, we have only to multiply the
area of the section by the velocity of the stream.
24. When the quantity of water Q discharged over a weir, is
known, the depth of the edge of the wasteboard, or H, may be
approximated from the following formulae, / length of wasteboard.
Q = 115/H^ nearly (XX.)
or, more accurately by adding the correction in article 15,
Q = ii5(^ + 4) «* (^Xl)
26. The quantities discharged for any given width are as the
^ power of the depth, or as H *.
Hence, to extend the use of the table to greater depths, wc have
only for
Twice any depth, take Q x 2*828
3 times Q x 5196
4 times Q x 8*000
5 times Q x 1M80
6 times Q x 14697
7 times Q x 18*520
8 times Q x 22*627
9 times Q x 27000
10 times Q x 31623
and the results will be nearly true.
To make them still more correct, where great accuracy is required,
add to them their thotisandth part.
Easamples of the use of* the Tables and Rules.
Ex, 1. Let the depth of a weir be 10 inches below the upper
surface of the water, and the width 8 inches. How many cubic feet
of water will be discharged in a minute ?
cub.fiB«t
By table § 23. Q for 10 inches deep and 1 inch wide = 13*535
Multiply this by / == 8
106280
Add ^>jy of this product 5*314
Discharge in one minute = 111*594
CHAP. V.j STRBAli liSASUKSBt. 97d
Em. S. Let the depth be 9 feet, and the width 1 fooL Required
the cubic feet discharged in a minute.
Bj the table Q for 12 inches deep and 1 inch wide s 17*805
Faetor for 9 times depth ss 27
Quantity for 1 inch width 480*735
Multiply by / » 12
5768802
Add ^9 of the product 266*441
Total quantity in cubic feet s= 6055*261
Ex, 3. Let a square orifice of 6 inches each side be placed in a
sluicegate with its top 4 feet below the upper surface of the water :
how much will it discharge in a minute ?
Here the quantity discharged by a slit in depth 48 inches, must be
taken from one in depth 54 inches.
cubic feet.
For 54, multiply Q for 6 by 3' or 27 169*965
For 48, Q for 12 by 4^ or 8 142440
Difference 27*525
S7*525 X (6 + ^) as 173*4 cubic feet, quantity discharged.
i^(C^.<— In an example like this, it is a good approximation, to
9nuUiyUf confinuaify together the area of the orifice^ the number 336*,
€^n£l ike equare root of the depth infect of the middle of the orifice.
Thus, in the preceding example, it will be  x ^ x 336 x >/4*25
3= ^ X 336 X 2*062 = 173*2 cubic feet.
The less the height of the orifice compared with its depth under
^e water, the nearer will the result thus obtained approach to
tte truth.
If the height of the orifice be such as to require consideration, the
2Mnciple off 7, page 261, may be blended with this rule.
Thnsy 9km\jmg ^is rule to Ex. 2, we shall have area x s/ depth
^ dd6 X I ss 9 X 3 X 224 = 6048, for the cubic feet discharged.
^niis is less than the former result by about its 900th part. It is,
ttlierefore, a good approximation, considering its simplicity: it may in
lEKianj VMes supersede the necessity of recurrence to tables.
Skcy. III. Coniriveaices to measure the velocity of running waters.
1. For these purposes, various contrivances have been proposed, of
^hieh two or three may be here described.
836 « 6 6 x 60.
274 STREAM MBASCREBS. [PART II.
Suppose it be the Telocity of the water of a river that is required ;
or, indeed, both the Telocity and the quantity which flows down it in
a given time. ObserTe a place where the banks of the nTer are
steep and nearly parallel, so as to make a kind of trough for the
water to run through, and by taking the depth at Tarious places in
crossing make a true section of the nTer. Stretch a string at right
angles OTer it, and at a small distance another, parallel to the first
Then take an apple, an orange, or other small ball, just so much
lighter than water as to swim in it, or a pint or quart bottle partly
filled with water, and throw it into the water aboTO the strings.
ObserTe when it comes under the first string, by means of a quarter
second pendulum, a stop watch, or any other proper instrument;
and obsenre likewise when it arrives at the second string. By
this means the Telocity of the upper surface, which in practice
may frequently be taken for that of the whole, will be obtained.
And the section of the riTer at the second string must be ascer
tained by taking various depths, as before. If this section be the
same as the former, it may be taken for the mean section: if not,
add both together, and take half the sum for the mean section.
Then the area of the mean section in square feet being muldplied
by the distance between the strings in feet, will give the contents
of the ^'ater in solid feet which passed from one string to the
other during the time of observation : and this, by the rule of three,
may be adapted to any other portion of time. The operation may
often be greatly abridged by taking notice of the arrival of the float
ing body opposite two stations on the shore, especially when it is not
convenient to stretch a string across. An arch of a bridge is a good
station for an experiment of this kind, because it affords a very regular
section and two fixed points of observation ; and in some instances
the sea practice of heaving the log may be advantageous. Where a
timepiece is not at hand, the observer may easily construct a quarter
second or other pendulum, by means of the rules and table relating
to pendulums in the Dynamics.
2. M. Pitot invented a stream measurer of a simple construc
tion, by means of which the velocity of any part of a stream may
readily be found. This instrument is composed of two long tabes of""
glass, open at both ends; one of these tubes is cylindrical throughout; z
the other has one of its extremities bent into nearly a right angle, ^
and gradually enlarges like a funnel, or the mouth of a trumpet:^
these tubes are both fixed in grooTes in a triangular prism of wood,^
so that their lower extremities are both on the same level, standings
thus one by the side of the other, and tolerably well preserred from^
accidents. The frame in which these tubes stand is graduated, closed
by the side of them, into dirisions of inches and lines.
To use this instrument, plunge it perpendicularly into the watery
in such manner that the opening of the funnel at the bottom of ontfS
of the tubes shall be completely opposed to the direction of the cur— 
rent, and the water pass freely through the funnel up into the tubei^
Then observe to what height the water rises in each tube, and note^
CHAP, v.] 8TRBAM MBA8UREB8. 275
the difference of the sides, for this difference will be the height due
to the velocity of the stream. It is manifest, that the water in the
cylindrical tuhe will be raised to the same height as the surface of
the stream by the hydrostatic pressure, while the water entering
from the current by the funnel into the other tube will be compelled
to rise aboTe that surface by a space at which it will be sustained by
the impulse of the moring fluid: that is, the momentum of the stream
iriU be in equilibrio with the column of water sustained in one tube
fhore the surface of that in the other. In estimating the velocity by
means of this instrument, we must have recourse to theory, as cor
rected by experiments. Thus, if A, the height of the column sus
tained by the stream, or the difference of heights in the two tubes, be
in feet, we shall have e = 6'5 v^ A, nearly, the velocity, per second,
of the stream; if A be in inches, then o = 22*47 %/ ^i nearly: or
&rther experiments made with the same instrument may a little
modify these coefficients.
It win be easy to put the funnel into the most rapid part of the
stream, if it be moved about to different places until the difference
of altitade in the two tubes becomes the greatest. In some cases it
will happen, that the immersion of the instrument will produce a
little eddy in the water, and thus disturb the accuracy of the observ
«tioii : but keeping the instrument immersed only a few seconds
will correct this. The wind would also affect the accuracy of the
experiments ; it is, therefore, advisable to make them where there is
little or no wind. By means of this instrument a great number
of curious and useful observations may easily be made: the velo
city of wator at various depths in a canal or river may be found
with tolerable accuracy, and a mean of the whole drawn, or they
maj be applied to the correcting of the theory of waters running
down gentle slopes. The observations may likewise be applied to
nmeertBon whether the augmentations of the velocities are in propor
tion to the increase of water passing along the same canal, or what
other relations subsist between them, &c.
Where great accuracy is not required, the tube with the funnel
%t hottom will alone be sufficient, as the surface of the water will be
Indicated with tolerable precision bv that part of the prismatic
fimme for the tube which has been moistened by the immersion : and
the wdocUks may be marked against those altitudes in the tube which
indicate them.
d. Another good and simple method of measuring the velocity of
water in a canal, river, &c., is that described by the Abbe Mann^ in
Itis treaitise on rivers; it is this: — Take a cylindrical piece of dry light
^wood, and of a length something less than the depth of the water in
the river ; about one end of it let there be suspended as many small
weights as may keep the cylinder in a vertical or upright position,
with ita head^just above water. To the centre of this end fix a
•mall straight rod, precisely in the direction of the cylinder's axis, in
order that, when the instrument is suspended in the water, the devia*
T 9
276 STRSAM MKA8UREBS. [l
tions of the rod from a perpendicularity to the mvrfmce of
indicate which end of the cylinder goes foremost, by wbieh
discovered the different velocities of the water at different
for when the rod inclines forward, according to the direction
current, it is a proof that the surface of the water has the
velocity ; but when it reclines backward, it shows that the
current is at the bottom ; and when it remains perpendiculai
sign that the velocities at the top and bottom are equal.
This instrument, being placed in the current of a river o
receives all the percussions of the water throughout the whol
and will have an equal velocity with that of the whole carre
the surface to the bottom at the place where it is pat :
by that means may be found, both with exactness and e
mean velocity of that part of the river for any determinate
and time.
But, to obtain the mean velocity of the whole section of tl
the instrument must be put successively both in the mid
towards the sides, because the velocities at those places are of
different from each other. Having by these means found the
velocities, from the spaces run over in certain times, the aritl
Tuean proportional of all these trials, which is found by divi<
common sum of them all by the number of the trials, will
mean velocity of the river or canal. And if this medium vel
multiplied by the area of the transverse section of the waten
place, the product will be the quantity running through that ;
a second of time.
The cylinder may be easily guided into that part which ^
to measure, by means of two threads or small cords, wh
persons, one on each side of the canal or river, must hold and
taking care at the same time neither to retard nor accelei
motion of the instrument.
4. Let A A' B B' be a hollow cylinder, open at both ends,
it be capable of being fixed by the side of a platform or of a
that its lower extremity BB^ may be placed at
any proposed depth below H R, the upper surface
of the stream. Let P P^ be pulleys, fixed at
opposite sides of the top and bottom of the tube.
To O, a globe of specific gravity nearly the same
as that of water, let a cord OP' PS be attached,
passing freely over the pulleys P', P, and having
sufficient length towards S to allow of its running
off to any convenient distance. Then, the bottom
of the tube being immersed to any proposed
depth, let the globe G be exposed to the free operation
stream; and as it is carried along with it, it will in 1, 2, i
seconds, or any other interval of time, drew off from a fizei
as S, a portion of cord; from which and the time elapsed, the
at the assigned depth will become known.
] STBBAM MBA8URBB8. 277
mvented, in 1720, an instrument called the Marine
'y for the doable purpose of measuring a ship's way, and
ing the velocities of streams. It is described in the Phil.
oL dd ; and in the succeeding volume a curious example of
given in ^ tables showing the strength and ffradual increase
aw of the tides of flood and ebb in the nver Thames, as
in Lambeth Reach." They are too extensive to be inserted
are very interesting, and may be seen in the Philosophical
ms AMdptedy vol. vii. p. 133.
378 EQUILIBRIUM OF ILASTIC FLUIDS. [PABT II.
CHAP. VI.
PNEUMATICS.
Sect. I. Weight and Equilibrium of Air and EUutic Fluids.
1. The fundamental propositions that belong to hydrostatics are
common to the compressible and incompressible fluids, and need not,
therefore, be repeated here.
2. Atmospheric air is the best known of the elastic fluids, and has ^
been defined to be an elastic fluid, having weight, and resisting com*^ ^
pression with forces that are directly as its density, of inversely as ^
the spaces within which the same quantity of it is contained.
The correctness of this definition is confirmed by experiment.
3. The weight of air is known from the Torricellian experiment, ^^^
or that of the barometer. The air presses on the orifice of the ^^.
inverted tube with a force just equal to the weight of the column of ~% ^
mercury sustained in it.
The weight of a cubic foot of air, with the barometer at 30 inches, ^ r^
and the temperature ^5^ of Fahr., is 1*2 ounces avoirdupois, or^^3
about v^rd part of the weight of a cubic foot of water under the — ^ i<
same circumstances.
A bottle, weighed when filled with air, is found heavier than after^r ^i
the air is extracted. The mean pressure of the atmosphere at^ ^t
London is 14*18 lbs. on every square inch of the earth's surface^p^^i
which is equivalent to the pressure of a column of quicksilver 28*89^^^ ^
inches in height; or a column of water nearly 33 feet in height— =^^*
Hence the total pressure on the convex surface of the earth amount^^ ^^
to 10,686,000,000 hundreds of millions of pounds.
The elastic force of the air is proved, by simply inverting a vesseT ^^*]
full of air in water : the resistance it offers to farther immersion, an^^ •^
the height to which the water ascends within it, in proportioira^^
as it is farther immersed, are proofs of the elasticity of the aixi^'C'^
contained in it*.
* It is in virtue of this property, and ought to be known as ottensivdy i
possible, that a man*s hat will serve in most cases as a temnormry lifepreMrva
to persons in hazard of drowning, by attending to the following direcdoos :
When a person finds himself in, or about to be in, the water, let him lay hok^ ^^
of his hat between his hands, laying the crown dose under his chin, and th^^ ^
mouth under the water. By this means, the quantity of air conuined in th^' ^
cavity of the hat will be sufficient to keep the head above water for icvtii^^ '
hoars, or until assistance can be rendered.
i
CHAP. VI.] WBIOHT AND EQUILIBRIUM OP ELASTIC FLUIDS. 279
4. When air is confined in a bent tube, and loaded with diflferent
weights of mercury, the spaces it is compressed into are found to be
inversely as those weights. But those weights are the measures of
the elasticity; therefore the elasticities are inversely as the spaces
which the air occupies.
The densities are also inversely as those spaces; therefore the
elasticity of air is directly as its density. This law was first proved
by Manotte's experiments. '
In all this, the temperature is supposed to remain unchanged.^
These properties seem to be common to all elastic fluids.
Air resists compression equally in all directions. No limit can be
assigned to the space which a given quantity of air would occupy if
all compression were removed.
5. In ascending from the surface of the earth, the density of the
air necessarily diminishes : for each stratum of air is compressed only
by the weight of those above it ; the upper strata are therefore less
compressed, and of course less dense than those below them.
6. Supposing the same temperature to be difl^used through the
atmosphere, if the heights from the surface be taken increasing in
arithmetical progression, the densities of the strata of air will decrease
in geometrical progression. Also, since the densities are as the com
pressing forces, that is, as the columns of mercury in the barometer,
the heishts from the surface being taken in arithmetical progression,
the columns of mercury in the barometer at those heignts will
decrease in geometrical progression.
7. Logarithms have, relatively to the numbers which they re
present, the same property, therefore if b be the column of mer
cury in the barometer at the surface, and at any height h above
the surface, taking m for a constant coefficient, to be determined
by experiment,
A = in(log6 — logg), or A = mlog (I.)
where m may be determined by finding trigonoroetrically the value
of A in any case, where b and $ have been already ascertained.
8. If ^ be the height of the mercury in the barometer at the
lowest station, fi at the highest, t and f the temperatures of the air at
those stations, f the fixed temperature at which no correction is
required for the temperature of the air; and if g and / be the
temperatures of the quicksilver in the two barometers, and n the ex
pansion of a column of quicksilver, of which the length is 1, for each
degree of heat; k being the perpendicular height (in fathoms) of the
one station above the other,
il = 10000(1 + 00244/^'^ A log —^ 77^] (II.)
« being nearly = j^.
280 ATMOSPHBRIO ALTlTUDli. [PABT 11.
If the oentigrade thermometer it used, hecaase the hegtHning of
the scale agrees with the temperature f^ so that /*« 0» the formula
hecomes more simple ; and if the expansion for air and nercurj he
hoth adapted to the degrees of thitf scale,
*= ,0000{. + 0044l(l±i:) log __J__^} (HI.)
9< The temperature of the air diminishes on ascending into the
Ittmosphere, hoth on account of the greater distance from the earth,
the principal source of its heat, and the greater power of aheorbing
heat that air acquires by being less compressed.
10. Professor Leslie, in the notes on his EkmenU qf Geometry^
p. 495 (edit. 2nd), has given a formula for determining the tempera
ture of any stratum of air when the height of the mercury in the
barometer is given. Tlie column of mercury at the lower of two
stations being &, and at the upper /?, the diminution of heat, in
degrees of the centigrade thermometer, is f  — T f 2^* Which
seems to agree well with observation.
1 1 . If the atmosphere were reduced to a body of the same density
which it has at the surface of the earth, and of the same temperature,
the height to which this homogeneous atmosphere would extend is,
in fathoms, equal to 4343 ^1 + '00441 ——V or, taking the
expansion according to Laplace, = 4343 ( 1 + ttt^ ) • • • C^^*)
Hence if h be the height of the mercury in the barometer, reduced
to the temperature /, the specific gravity of mercury is to that of airi
as ft to 4343 ( I + Tjwjj: J ^ or the specific gravity of air
h
72 X 4343 X (\ ^ ~\ '
\ ^ 1000/
The divisor 72 is introduced in consequence of b being expressed
in inches.— (jP/tfj^atV^ Outiines.)
12. The^ Telocity with which air rushes into a vacuum is equal
to that which a heavy body would acquire in Mling from a height
equal to that of a homogeneous atmosphere equivalent in wemit
to that of the air at the time. Thus if H be the height of Uie
homogeneous atmosphere, deduced from the formula (IV.) abote,
expressed in feet, and V the velocity of the air in feet per second^
Wd have from the formula, page 215,
V = S <V.)
CHAP. VI.] ATM08PHBBIC ALTITUDES. 281
From wbieh we find the value of V for the mean temperature and
preasare to be 1860 feet per second.
13. The velocity with which sound traverses the atmosphere de
pends not only upon the direction and force of the wind, but also on
the temperature and density of the air at the time, and the quantity
of moisture which it contuns. In the following formula given by
Mr. Galbraitb, of Edinburgh, the whole of these circumstances are
taken into account Let v be the velocity of sound in feet per
second, t the temperature of the air, 8 the elastic force of vapour for
the due point, the barometric pressure, x the latitude of the place
of observation, v the velocity of the wind, and ^ the angle which the
direetion of the wind makes with that of the sound ; then
»a= (1024225 + 01103 0(l + ^i qZ. 2e ) C^^'^'''^®
— 001378. cos 2 a) H w. cos ^ (VI.)
8bct. II. Madtine$ for BaUing Water by the Pressure of the
Atmoephere.
1. The term Pump is generally applied to a machine for raising
water by means of the air's pressure.
2. The common suctionpump consists of two hollow cylinders,
which have the same axis, and are joined in A C (fig. 216). The lower
is partly immersed, perpendicularly, in a spring or reservoir, and is
called the suction^ube; the upper the body of the pump. At AC is a
fixed sucker containing a valve which opens upwards, and is less than
34 feet from the surface of the water. In the body of the pump is
a piston D made airtight, movable by a rod and handle, and con
taining a valve opening upwards. And a spout G is placed at a
greater or less distance, as convenience may require, above the greatest
elevation of D.
The action of this pump is as follows. Suppose the movable pis
ton D at its lowest depression, the cylinders free from water, and the
air in its natural state. On raising this piston, the pressure of the
air above it keeping its valve closed, the air in the lower cylinder A B
forces open the valve at A C, and occupies a larger space, viz., be
tween B, the surface of the water, and D ; its elastic force, therefore,
being diminished, and no longer able to sustain the pressure of the
external air, this latter forces up a portion of the water into the cylin
der AB to restore the equilibrium. This continues till the piston has
reached its greatest elevation, when the valve at A C closes. In its
subeeqaent descent, the air below D becoming condensed, keeps the
valve at AC closed, and escapes by forcing open that at D till the
piston has reached its greatest depression. In the following turns a
similar effect is produced, till at length the water rising in the cylin
der forces open the valve at A C, and enters the body of the pump ;
282 PUMPS. [part II.
when, by the descent of D, the valve in AC is kept closed, and the
water rises through that in D, which on reascending carries it for
ward, and throws it out at the spout O.
3. Cor, 1. The greatest height to which the water can be raised in
the common pump by a single sucker is when the column is in eqoi
Hbrio with the weight of the atmosphere, that is, between 32 and 36
feet.
4. Cor. 2. The quantity of water discharged in a given time is de
termined by considering that at each stroke of the piston a quantity
is discharged equal to a cylinder whose base is a section of the pump,
and altitude the play of the piston.
5. To determine the force necessary to overcome the resistance
experienced by the piston in ascending. Let h = the height H F (fig.
217) of the surface of the water in the body of the pump above EF,
the level of the reservoir; and a^ = the area of the section MN.
Let h I ^ the height of the column of water equivalent to the pres
sure of the atmosphere ; and suppose the piston in ascending to arrive
at any position mn which corresponds to the height IF. It is evi
dent that the piston is acted upon downwards by the pressure of the
atmosphere ^ d^ h.^ and by the pressure of the column B m = er
X H I ; therefore the whole tendency of the piston to descend = a*
(A + HI).
But the piston is acted upon upwards by the pressure of the air on
the external surface £ F of the reservoir = c^ hy\ part of which is
destroyed by the weight of the column of water having for its base
mn, and height FI;
.*. the whole action upwards = a^ x (^j — FI);
whence F = a* . (A, f HI) = 0^.(^1— FI)
= a«. FH == d'h,
that is, the piston throughout its ascent is opposed by a force equal to
the teeight of a column of water having the same hose as the piston^
and an altitude equal to that of the surface of the water in the bodjf of
the pump above that in the reservoir. In order, therefore, to produce
the upward motion of the piston, a force must be employed equal to
that determined above, together with the weight of the piston and
rod, and the resistance which the piston may experience in conse
quence of the friction against the inner surface of the tube ♦.
* Suppose the body of the pump to be 6 inches in diameter, and the greatest
height to which the water is raised to be 30 feet ; suppose, also, the weight of
the niston and its rod to be 10 lbs., and the friction onefifUi of the whole
weight. Then, ^ of the square of the diameter gives the ale gallons in
a yard in length of the cylinder, and an ale gallon weighs 10 Iba. Therefore
(6< X 10) + % (6* X 10) »= 360 h 74 " 367*4 lbs., weight of the opposing
column of water. And 367 4 + 10 f i (377*4) = 452*9 lbs., whole opposing
pressure.
If the piston rod be moved by a lever whose arms are as 10 to 1, this pres
snre will be balanced by a force of 46*29 lbs., and overcome by any greater
force.
CHAP. VI.] SUCTION PUMP. 383
When the piston begins to descend, it will descend by its own
weight ; the onhr resistance it meets with being friction, and a slight
impact against the water.
6. Cor. 1. If the water has not reached the piston, let its level
be in e ir. The under surface of the piston will be pressed by the
mtemal rarefied air. But this air, together with the oolumn of
water, £ o, is in equilibrio with the pressure of the atmosphere a' A, ;
and . • its pressure =s a' . (Ai — E 9). And the pressure downwards
= «•*.;
Hence the force requisite to keep the piston in equilibrio increases
as the water rises, and becomes constant and = a'A as soon as the
water reaches the constant level B H.
7 Cor, 2. If the weight of the piston be taken into the account,
let this weight be equal to that of a column of water whose base is
m II and height/), ss a*/? ;
.. F = (i'. (Er+/?).
8. To determine the height to which the virater will rise after one
stroke of the piston ; the fixed valve being placed at the junction of
the suctiontube and body of the pump : supposing that afler eyery
elevation of the piston there is an equilibrium between the pressure
of the atmosphere on the surface of the water in the reservoir, and
the elastic force of the rarefied air between the piston and surface of
the column of water in the tube, together with the weight of that
column.
Let ah {j^g, 218) be the surface of the water in the suctiontube,
after the first stroke of the piston : if the piston were for an instant
stationary at D, the pressure of the atmosphere would balance E 6,
and the elastic force of the air in N a.
Let A E, the height of the suctiontube, = a,
D R, the play of the piston, == &,
h s: the height of a column of water equivalent to the pressure of
the atmosphere,
y ^ the height of a column equivalent to the pressure of the air
in No,
m =E Eo,
^ =r 3*14159
and B and r = the radii of the body and the suction tube.
Then « + jr = A,
AF vHa hf^a
I hr'* a
whence A = ^0 + ^ , . , r ;
.. AR«6 + hf*a — hr*a = R**« + r*a« — r*a?» f ht^a.
*y^* ^ reduced vj^^^li^^rf^^
^.«. these arc ^^ ^^^^^^^^Tii^^^^.
^^ —  — TnX.
«A BO on. ,^ " 1 ^ »8ceot8' ?fc+« —*•" '
rise aftet^'/tVve «'**'"
CHAF. VI.] SUCTION PUMP. 285
Uon due to each particular stroke, the differences of those elevations,
and the successive differences in the elastic force of the remaining air,
may be known.
10. If the weight of the valve c be not considered, it is evident
that after a certain number of strokes a vacuum will be produced in
the suctiontube, provided it be equal to or not greater than the
height due to the pressure of the atmosphere, that is, if a be not
greater than h.
For, in this case, x^ =. x^.,,
•nd..«^,= J{jt,— y^T* — 4Am*4a?. . . (A f o — «.i)}, (IX.)
whence «»., = ky the greatest height of the column of water in the
tube. If, therefore, the length of the suctiontube do not exceed the
height due to the pressure of the atmosphere, the water will continue
to ascend in it after every stroke of the piston, till at length it will
into the body of the pump.
But if the altitude of A F be greater than k, the water will con
tinue to ascend without ever reaching its maximum height. For, in
this case, an actual vacuum cannot be produced ; and as ^^ f y* = ^'*
and y, can never become = 0; .*. ^« can never = ^*. But, tlie
successive values of ^r continually decreasing, the corresponding values
of as will continually increase.
11. If the weight of the valve c be taken into the account, a
column of water must be added equal to the additional pressure to be
overcome. Let / = the height of this column, then
a + y^l^h;
and .'. a + y ^ h ^ I = hy
If therefore this value of A be substituted for A, the preceding
equations are applicable.
12. In the preceding cases, the movable piston has been supposed
to descend to A C. If it does not, it may happen that the water may
not reach AC, though AC be less than' 34 feet from the surface of
the water in the reservoir.
After the first elevation of the movable piston to its greatest alti
tude, c being closed, the elastic force of the air between DN and AC
is (A — «), and its magnitude x h R*. If, in descending, the piston
describes a space ^ less than ^, so as to stop at a distance b ^ b*
from AC, this magnitude becomes {b  b") . v B,^ ; .*. the elastic force
is (A — «) . Y 7T. Now in order that the pressure upwards may
open the valve, this must exceed the elastic force of the atmosphere ;
* Hence it appears that it is not ttrieUy true, that wat«r will ascend In the
•actiontube to a height equal that of a column equivalent to the pressure of
the atmoephere. This is a limit to which it approximates, but does not reach
in a finite time.
286 SUCTION PUMP. [part II,
SUCTION
PUHP.
..(*«),
h
or
(A  *) . i
>h.{hb');
.:hx<Lhh',
X V
If .% Tbe less than , the valve DN will Dot open; there
h
will therefore be the same quantity of air between A C and the
sucker: which, when the piston has reached its highest elevation,
will have the same elastic force as that between A C and a' h' ; and
therefore c, being equally pressed on both sides, will remain un
moved, and the water will not ascend.
13. If the fixed valve be placed at the surface of the water; to
determine the ascent of the water in the suctiontube.
Let £ a, £ a^ be the successive heights to which the water rises ;
then, after the first ascent of the piston,
mo f a — «
whence a? =: J . {/? — Jp^ — ^hmh}
and y = i . {2k — p f >//?* — 4tkmh}^
which equations are the same as were determined for the first ascent
of the piston (§ 8). Therefore, in the same manner as before,
we shall have «, = J . {/? — s/p^ — 4>hmh — 4Aa?,.,},
14. If the water be supposed to stop after (n f IX ascents of the
piston, then «, = a?»_, ;
and .*. d?,_, = i . {/> — "^p^ — ^hmh — 4Aa?^,},
whence d?,_i = J . {a f mh ± >/(a + mhY — ihmb} ...(X.)
Hence, therefore, there are two altitudes at which the water may
stop in its ascent, if (a + mbY is equal to or greater than 4 Am 5. In
the former case the two values of x^^^ are equal, that is, there will
be only one altitude = ^ . (a + m 6), at which the water will stop.
In the latter case there are two which may be ascertained.
If 4 Am* be greater than (a + nthf^ the water will not atop.
Ex.l. If A = 32 feet, a = 20, 6 = 4, and m = 1, or the
suctiontube and body of the pump be of the same diameter.
«— I = I . {20 + 4 ± V(24)«  4 . 1 . 32 . 4} = i . {24 ± V64}
= le or 8.
CHAP. VI.] VORCINO PUMP. 287
Ex. 2. If A = 32 feet, a = 25, ft = 2, and wi = 4,
x^. = 1 . {25 H 8 ± >/(33)' — 4 . 32 . 4 . 2} = J . {33 ±
>/65} = 418062 or 241938.
15. If m = 1, or the tubes have the same diameter,
«. = i . {« + ft ± >/(« + hf  4A6},
which is imaginary, if (a + 5)^ is less than 4 A ft, or ft greater than
(o + fty
4A '
In order, therefore, that this pnmp may produce its effect, the play
of the piston muit be greater than the sqtiare of its greatest akitttde
above Vie surface of the water in the reservoir divided by four times
the height due to the pressure of the atmosphere,
16. The lifingpump consists of a hollow cylinder, the body of
which is immersed in the reservoir. It is furnished with a movable
piston, which, entering below, lifts the water up, and is movable by
means of a frame which is made to ascend and descend by a handle.
The piston is furnished with a valve opening upwards. A little
below the surface of the water is a fixed sucker with a valve opening
upwards. This is an inconvenient construction, upon the peculiari
ties of which we need not dwell.
17 The forein^'pump consists of a suctiontube AEFC (fig. 219)
partly immersed m the reservoir, of the body of the pump ABOC,
and of the ascending tube HK. The body is furnished with a
movable solid sucker or plunger, D, made airtight. And at AC
and U are fixed suckers with valves opening upwardd.
18. To explain the action of this pnmp. Suppose the plunger D
at its greatest depression ; the valves closed, and the air in its natural
state. Upon the ascent of D, the air in A CD occupying a greater
space, its elasticity will be diminished, and consequently the greater
elasticity of the air in A F will open the valve at A C, whilst that at
H is kept closed by the elasticity of the external air ; water there
fore will rise In the suction tube. On the descent of D from its
greatest elevation, the elasticity of the air in the body of the pump
will keep the valve A C closed, and open that at H, whence air will
escape. By subsequent ascents of the piston, the air will be ex
pelled, and water rise into the body. The descending piston will
then press the water through the valve at H, which will close, and
prevent its return into the bodv of the pump ; D therefore ascending
again, the sp^ce left void will be filled by water pressing through the
valve AC; and this upon the next ascent of D is forced into the
ascending tube ; and thus, by the ascents and descents of D, water
may be raised to the required height.
19. In this pump D must not ascend higher than about 32 feet
from the surface of the water in the reservoir.
20. To determme the force necessary to overcome the resistance
experienced by the piston : —
288 FIRE INOINB. [past II.
Let h = the height of a column of water equivalent to the presiare
of the atmosphere, and E B the height to which the water is forced.
Let M N he any position of the piston D whose area = A, and the
weight of the piston and its appendages = P. Let X = the force
necessary to push the piston upwards during the suction, friction not
heing considered, and V s= that employed to force it down.
When the piston ascends, and H is closed
X = P + AA — A.(A — ME)
= P f A . ME.
Let the sucker he in the same position in its descent, and therefore
A C closed, and H open,
Y = AA fA.MB — (AA f P)
= A . M B — P.
Hence X + Y = A.EB; or the whole force exerted, in the case
of equilihrium, is equal to the weight of a column of water whose
base is equal to that of the piston, and altitude the distance between
the surface of the water and the point to which it is to be raised.
21. In this pump the effort is divided into two parts, one opposed
to the suction, and the other to the forcing ; whereby an advantage
is gained over the other pumps where the whole force is exerted at
once whilst the water is raised.
22. In order to have the force applied uniform, lot X = Y ;
.. PA.ME==A.MB — P;
.. P = JA . (MB — ME).
The piston therefore must play in such a manner that M B may be 
greater than M E.
23. In the common forcing pump, the stream b intermitting ; for '
there is no force impelling it during the return of the sucker.
One mode of remedying this, is by making an interruption in the
ascending tube, which is surrounded by an air vessel T (fig. 220);
in which, when the water has risen above Z, the air above it is com
pcssed, and by its elasticity forces the water up through Z; the
orifice of which is narrower than that of the tube, and therefore the
quantity of water introduced during the descent of the piston will
supply its discharge for the whole time of the stroke, producing a
continued stream.
24. The fireengine consists of a large receiver A B C D, called the
airvessel, into which water is driven by two forcingpumps EF,
OH (fig. 221), (whose pistons are Q an^ R), communicating with
its lower extremities at I and K, through two valves opening
iniiards. From the receiver proceeds a tube M L through which
the water is thrown, and directed to any point by means of a pipe
movable about the extremity L. The pumps are worked by a
lever, so that whilst one piston descends the other ascendi. The
pumps communicate with a reservoir of water at N.
CHAP. VI.] QUICKSILVER PUMP. 289
25. To explain the action of this engine.
The tube N being immersed iu the reservoir, and the piston R
drawn up, the pump OH becomes filled; and the descent of the
piston R will, as in the forcingpump (§ 1 8), keep the valve H close,
and cause the water to pass into the airvessel by the valve I, whilst,
by the weight of the water in the airvessel, the valve K will be
kept abut. In the same manner, when R ascends, Q descending
will force the water through K into the airvessel. By this means
the air above the surface of the water becoming greatly compresse<1
will, by its elasticity, force the water to ascend through M L, and to
issue with a great velocity from the pipe*.
26, When the airvessel is half full of water, the air being then
compressed into half its natural space will have an elastic force
equal to twice the pressure of the atmosphere : therefore, when the
stopoock is turned, the air within pressing on the subjacent water
with twice the force of the external air, will cause the water to spout
from the engine to the height of (2 — 1)83, or 33 feet; except so
far as it is diminished by friction.
Or, genemlly, if denote the fractional height of the water in
II
the airbarrel, then  will denote the height of the space occupied by
the compressed air, n times the pressure of the atmosphere its elastic
force, and (» — 1 ) 33, the height in feet to which the water may be
projected.
Thus, if I of the airbarrel be the height of the water, the elastic
force of the air will be four times the pressure of the atmosphere,
and (4 — 1) 33 =s 99 feet, the height to which the water may then
be thrown by the engine.
27 The modifications in the constructions of pumps with a view
to their practical applications are very numerous. Those who wish
to acquaint themselves with some of the most useful, may consult
the 2nd Tolume of my Treatise on MechanicSy and Nos. 13, 41, 69,
and 93, of the Meehaniee Magazine,
Id addition to these, there may now be presented a short account of
uauieksiivervump^ which has been recently invented by Mr. Thomas
Clark of Edmburgh, and which works almost without friction. It
has great power in dra\nng and forcing water to any height, and is
extremely simple in its construction. In fig. 222, a a is the main
pipe inserted into the well b ; a valve is situated at c, and another at
dj both opening upwards ; a piece of iron tube is then bent into a
drcular form, as at/^ again turned off at g in an angular direction, so
as to pass through a stuffing box at A, and from thence bent out
wards as at t, connecting itself with the ring. A quantity of quick
* Tbe preceding part of this section Is taken from Bland*s Hydrottatics ; a
nxj dqgant and valuable work, which I beg most cordially to recommend
to those who wish to obtain a oomprehenaive knowledge of the Uieory of this
■ ■ — at of adeaee.
290 SPIRAL PUMP. [part II.
silver is then put into tlie ring filling it from q to q\ and the ring
being made to vibrate upon its axis hy a vacuum is soon effected
in the main pipe by the recession of the mercury from p to q^
thereby causing the water to rise and fill the vacuum : upon the
motion being reversed, the quicksilver slides back to ^, forces up the
water and expels it at the spout e.
*' Mr. Clark calculates that a pump of this description with a ring
twelve feet in diameter, Hdll raise water the same height as the
common lifting pump, and force it one hundred and fifty feet higher
without any friction." {Mechanics' Register and Jamiesons Edin
burgh Journal,)
28. It is usual to class with pumps, the machine known by the s^
name o^ Archimedes* scretCy or the teaterstiail. This consists either imt
of a pipe wound spirally round a cylinder, or of one or more spiral J'.m^
excavations formed by means of spiral projections from an internal X^^a^
cylinder, covered by an external cylindrical case, so as to be wator^ r3cr
tight. The cylinder which carries the spiral is placed aslant, so as io^^M ^\o
be inclined to the horizon in an angle of from 30^ to 45°, and£» Mnd
capable of turning upon pivots in the direction of its axis posited atirwEx at
each extremity. The lower end of the spiral canal being immersedC» ^^ed
in the river or reservoir from which water is to be raised, th^ MrMiie
water descends at first in the said canal solely by its gravity; but «l.» *ut
the cylinder being turned, by human or other energy, the water '^^ ^er
moves on in the canal, and at length it issues at the upper extremitj^^ ^^7
of the tube.
Several circumstances tend to make this instrument imperfect ana» ^rsi^
inefficacious in its operation. The adjustments necessary to insure ^* ^
a maximum of effectual work are often difficult to accomplish. 1 ^^ "
seldom happens, therefore, that the measure of the work done ex— :^^"
ceeds a tliird of the power employed : so that this apparatus, not— ^^'*
withstanding its apjmrent ingenuity and simplicity, is very sj>aringljC— ^
introduced by our civil engineers.
29. Spiral pump. This machine is formed by a spiral pipe or ^'
several convolutions, arranged either in a single plane, as in fig. 223. >
or upon a cylindrical or conical surface, and revolving round an axis ^ *
The curved pipe is connected at its inner end, by a central water
tight joint, to an ascending pipe, r P, while the other end, S, receivec. .^
during each revolution, nearly equal quantities of air and water««i^
This apparatus is usually called the Zurich machine, because it wa^
invented, about 1746, by Andrew Wirtz^ an inhabitant of Zurich*^
It has been employed with great success at Florence, and in Russia^
and the late Dr. Thomas Young states, that he employed it advantage—
ously for raising water to a height of forty feet. The outer end of
the pipe is furnished with a spoon, S, which contains as much water
as will half fill one of its coils. The water enters the pipe a little before
the spoon has reached its highest position, the other half remaining
full of air. This air communicates the pressure of the column of
water to the preceding portion; and in this manner the effect of
nearly all the water in the wheel is united, and becomes capable of
CHAP. VI.] SCHEMNITZ VESSELS. 291
sapporting tlie column of water, or of water mixed with air, in the
ascending pipe. The air nearest the joint is compressed into a space
much smaller than that which it occapied at its entrance ; so that,
when the height is considerahle, it becomes advisable to admit a
larger portion of air than would naturally fill half the coil. This,
while it lessens the quantity of water raised, lessens also the force
requisite to turn the machine. The loss of power, supposing the
machine well constructed, arises only from the friction of the water
against the side of the pipes, and that of the wheel on its axis : and
where a large quantity is to be raised to a moderate height, both of
these sources of resistance may be rendered very inconsiderable.
30. Schemnitz vessels^ or the Hungarian machine. The media
tion of a portion of air is employed for raising water, not only in the
spird pump, but also in the airvessels of Schemnitz, as shown in
fig. 224. A column of water, descending through a pipe, C, into a
closed reservoir, B, containing air, obliges the air to act, by means of
a pipe, D, leading from the upper part of the airvessel, or reservoir,
on the water in a second reservoir. A, at any distance either above it
or below it, and forces this water to ascend through a third pipe, E,
to auj height less than that of the first column. The airvessel is
then emptied, the second reservoir filled, and the whole operation
repeated. The air, however, must acquire a density equivalent to
tlie requisite pressure before it can begin to act : so that, if the height
of the columns were thirtyfour feet, it must be reduced to half its
natural space before any water could be raised, and thus half of the
force would be lost. But where the height is small, the height lost
in this manner is not greater than what is usually spent in overcoming
friction, and other imperfections of the machinery employed. The
force of the tide, or of a river rising and falling with the tide, might
easily be applied to the purpose of raising water by a machine of this
kind. Thus, if at low tide the vessel A were filled with air, then, at
high tide, the water flowing down the tube £, would cause the water
in the vessel B to ascend in the pipe C.
31. The hydraulic ram. In this hydraulic arrangement, the
momeiitum of a stream of water flowing through a lon^ pipe is
employed to raise a small quantity of water to a considerable heiglit.
The passage of the pipe being stopped by a valve which is raised by
the stream, as soon as its motion becomes sufiiciently rapid, the
whole column of fluid must necessarily concentrate its action almost
instantaneously upon the valve. In these circumstances it may be
regarded as losing the characteristic property of hydraulic pressure,
and to act almost as though it were a single solid : so that, supposing
the pipe to be perfectly elastic and inextensible, the impulse may
overcome almost any pressure that may be opposed to it. If another
valve opens into a pipe leading to an airvessel, a certain quantity of
the water will be forced in, so as to condense the air, more or less
rapidly, to the degree that may be required for raising a portion of
the water contuned in it to a given height.
The late Mr. Whitehurst appears to have been the first who
u 2
292 FORCE OP THR WIND. [PART II.
employed this method: it was afterwards improved by Mr. Boulton.
But, like many English inyentione, it never was adequately estimated,
until it was brought into public notice by a Frenchman. M. Mont
golfier, its reinventor, gave to it the name wliich it now bears of the
Hjfdravlic Ram, in allusion to the battering ram.
The essential parts of this machine are represented in figure 225.
When the water in the pipe A B (moving in the direction of the
arrows) has acquired sufficient velocity, it raises the valve B, which
immediately sto])s its farther passage. The momentum which the
water has acquired then forces a portion of it through the valve, C,
into the airvessel, D. The condensed air in the upper part o( D
causes the water to rise into the pipe £, as long as the effect of the <^^ ^e
horizontal column continues. When the water becomes quiescent, ^..^.t
the valve B will open again by its own weight, and the current along ^^ «i
A B will be renewed, until it acquires force enougli to shut the said£» M^i
valve B, open C, and repeat the operation.
The motion in the horizontal tube arises from the acceleration oft^ <:^ o
the velocity of a liquid mass falling down another tube, and com^ .fliiKim
municating with this.
In an experiment made upon an hydraulic ram, at Avilly, n eii m^ ^jak
Sen lis, by M. Turquet, bleacher, the expense of power was found t*,:^ U
be to the produce, as 100 to 62. In anotlier, as 100 to 55; in tvr^^^^wi
others, as 1 00 to 57. So that a hydraulic ram placed in favourabi f ^Jblc
circumstances, may be reckoned to employ usefully rather more thi mwM^Mn
half its force.
*i^* For more full accounts of the three last contrivanoes, tlr:^.JRlie
reader may consuH the 2nd volume of my Mechanics,
Sect. III. Force of the Wind.
1 . Air, when in continuous motion in one direction, becomes ^^ss a
very useful agent of machinery, of greater or less energy, accordir .^iog
to the velocity with which it moves. Were it not for its vsrialMlfir Mlitj
in direction and force, and the consequent fluctuations in its auppT ^ "^ly?
scarcely any more appropriate fii'st mover could generally be wnijl^ ^""^
for. And even with all its irregularity, it is still so useful as to
require a separate consideration.
2. The force with which air strikes against a moving surfaoe,
with which the wind strikes against a quiescent surface, is nearly
the square of the velocity : or, more correctly, the exponent of i
velocity, determined according to the rule siven in Example
page 96, varies between 2*03 and 2*05; so Uiat, inmost practical/
cases, the exponent 2, or that of the square, may be employed wiC^
out fear of error. U $ he the angle of incidence, s the surface struct
in square feet, and v the velocity of the wind, in feet, par teoond;
then, if / equals the force in avoirdupois pounds, either of die two
following approximations may be used, viz. : —
CHAP. VI.] FORCB OP THE WIND.
293
/=
440
(XI.)
or,/= 002288 c'^wn'jg (XIT.)
Of these, the first is usually the easiest in operation, requiring only
two lines of short division, viz., by 40 and by 1 1 .
If the incidence be perpendicular, sin^/3 = 1, and these become,
/= ^ = 002288 «»« (XIII.)
3. The table in the margin exhibits the
Force of the wind when blowing perpen
dicularly upon a sur^Mse of one foot square,
at the several velocities stated. The velo
city of 80 miles per hour, is that by which
the aeronaut Gamerin was carried in his
tiaUoon from Ranelagh to Colchester, in
June, 1802. It was a strong and boister
ous wind, but did not assume the character
of a hurricane^ although a wind with that
velocity is so characterized in Rouse's table.
In Mr. Green's aerial voyage from Leeds,
in September, 1823, he travelled 43 miles
in 18 minutes, although his balloon rose to
the height of more than 4000 yards.
4. Borda found by experiment, in the year 1762, that the force
of the wind is very nearly as the square of the velocity, but he
iSBUDed that force to be ereater than what Rouse found (as expressed
in die above tahle) in tne ratio of 111 to 100. Borda ascertained
also^ as was natulfal to expect, that, upon difierent surfaces with the
same velocity, the force increased more rapidly than the surface.
M. Valz, applying ttie method of the minimum squai^es to Borda's
resoltSy ascertained tiiat the whole might be represented by the
formula
^ = 0001289 a^ + 0000030541 ar* (XIV.)
and nearly as correctly by
y = 000108 »" (XV.)
^ representing the surface in square inches (French), and p the
force oorrespondibg to the velocity of 10 feet per second expressed in
French pounds*
Velocity of the
Wfnd.
Perpendi
cular force
Milen
= feet
foot in
fa one
inane
•▼OlIttupOlB
pound*.
hour.
1
second.
147
•005
8
293
•020
3
440
•044
4
587
•079
5
733
•123
10
1407
•402
15
2200
I 107
20
20*34
1908
25
30*67
3075
30
4401
4420
35
51 34
0027
40
5868
7873
45
00 01
0063
60
7336
12300
00
88 02
17716
80
11730
31400
100
14670
40200
294 WAT£R AS A MECHANICAL AOBNT. [PAHT ^j^
CHAP. VII.
MECHANICAL AGENTS.
Sect. I. Water as a Mechanical Anient,
1. The impulse of a current of water, and sometimes its weight
and impulse jointly, are applied to give motion to mills for grinding
corn and for various other purposes. Sometimes the impulse is ap
plied ohiiquely to floatboards in a manner which may be compre
hended at once by reference to a amokeJack^ in which the asoendifiS
smoke strikes the vanes obliquely, and communicates a rotatory in*"
tion. If we imagine the wLole mechanism to be inverted, m^'^*^
to fall upon the vanes, rotation would evidently be produced ; m^^,
that with greater or less energy in proportion to the qoantitj ^
water and the height from which it falls. i
Waterwheels of this kind give motion to mills in Germany, a ^^ ^
some other parts of the Continent of Europe. I have also w""'^;
mills of the same construction in Balta, the northernmost Sbetla^^^
isle. But wherever they are to be found, they indicate a very vT^^
perfect acquaintance with practical mechanics; as they occasion
considerable loss of power.
2. Water frequently gives motion to mills, by means of what
technically denominated an undershot wheel. This has a number ^
planes disposed round its circumference, nearlv in the direction m ^
its radii, and these floathoards (as they are called) dipping into tl*^"
stream, are carried round by it, as shown in fig. 226. The axle c^
the wheel, of course, by "the intervention of proper wheels an' ^^
pinions, turns the machinery intended to be moved. Where tb ^^
stream is large and unconfined, the pressure on each floatboard is thi^^
corresponding to the head due to the relative velocities (or differenc^^^
between the velocities of stream and floatboard) : this pressare v^^
therefore, a maximum when the wheel is at rest; but the wwrkfer^'^
formed is then nothing. On the other hand, the pressure is noibin^^
when the velocity of the wheel equals that of the stream. Goose — "^
quently, there is a certain intermediate velocity, which CMises tb^^
work performed to be a maximum.
The weight equal to the pressure is Q (^/ H — ^ ^)', Q beinp
the quantity of water passing in a second, H the height due to V the
velocity of the water, and h that due to U the velocity of the floefr
board. Considering this as a mass attached to the wheel, its moTing
force is obtained by multiplying it into U : and as >/ H — ^^ A vmries
CHAP. VII.] WATBBMJLLS. 295
as V — U, this moviDg force varies as (V — U)^ . U which is a maxi"
mum when U = ^l V. In this case, then, the rim of the wheel moves
with ^ of the velocity of the stream ; and the effect which it pro
duces is
Q X (fV)« X 4V = ^QV»:
80 that the work performed hy an undershot wheel, according to the
usual theory, equals ^j of the moving force.
Friction, and the resistance of fluids, modify these results; hut
Smeaton and others have found that the maximum work is always
obtained when U is between ^ V and ^ V.
3. Where the floats are not totally immersed, the water is heaped
upon them ; and in this case the pressure is that due to 2 H.
4. When the floatboards move in a circular sweep close fitted to
them, or, in general, when the stream cannot escape >vithout ac
quiring the same velocity as the wheel, the circumstances on which
the investigation turns become analogous to what happens in the
collision of nonelastic bodies. The stream has the velocity V before
the stroke which is reduced to U, and the quantity of motion corre
sponding to the difference, or to V — U, is transferred to the wheel ;
this turns with the velocity U; and therefore the eflfcct of the wheel
(V— U\ VU— U* ,.,.
I U, or — ; which is a maximum when V ^
2 U ; being then ^ of the moving power.
Hence appears the utility of constraining the water to move in a
narrow channel.
5. The undershot wheel is used where a large quantity of water
can be obtained with a moderate fall. But where the fall is con
siderable the overshot (fig. 227) is almost always employed. Its cir
cumference is formed into angular buckets, into which the water is
delivered eithef at the top or within 60** of it : 52**  is the most
advantageous distance. In that case, if r = the full radius of the
wheel, H the whole, and h the effective height of the fall, A = r
(1 i sin 37*"^) = 1605 r, and r = '623 h. If the friction be
aboat f of the moving power, the velocity of the circumference of
the wheel to produce a maximum effect, will = 2*07 >/ H. Here,
too, a fall of ^ H will give the water its due velocity of impact upon
the wheel: and 122*176 s U^ equals the mechanical effect in
pmrndfl, 8 being the section, in feet, of the stream that supplies the
backets.
Mr. Smeaton's experiments led him to conclude that overshot
wheels do most work when their circumferences move at the rate of
3 feet in a second, and that when they move considerably slower than
this, they become unsteady and irregular in their motion. This de
termination is, however, to be understood with some latitude. He
mentions a wheel 24 feet in diameter, that seemed to produce nearly
ils fall effect though the circumference moved at the rnte of G feet in
a second ; and another of the diameter of 33 feet, of which the cir^
296 WATEBMILLS. [PABT H.
curafereiicc had only a velocity of 2 feet in a second, without any
considerable loss of power. The first wheel turned round in 12*6
seconds, the latter in 51*9 seconds.
0. Where the fall is too small for an overshot wheel, it is most
advisable to employ a breast^wheel (Plate VI.) which partakes of iu
{)roperties ; its floatboards meeting at an ansle, so as to be assimi
ated to buckets, and the water beins: considerably confined within
them by means of an arched channel fitting moderately close, but
not so as to produce unnecessary friction. But when the circum —
stances do not admit of a breast wheel, then recourse must be had to^:=a
the undershot. For such a wheel it is best that the floatboards be^^
so placed as to be perpendicular to the surface of the water at th^^ j
time they rise out of it ; that only one half of each should ever b»^^:3
below the surface, and that from 3 to 5 should be immersed at onc^»«^:
The Abbe Mann proposed that there should not bo more than six c^
eight floatboards on the whole circumference.
7. Mills moved by the reaction of water are usually denominate^K^
Barkers Mills; sometimes, however. Parent's; at others, Segner^ ^
But the invention is doubtless Dr. Barkers. Their construction i
shown in fig. 228, where C D is a vertical axis, moving on a pivot
D, and carrying the uj)per millstone m, after passing through e
opening in the fixed millstone C. Upon this axis is fixed a vertic^^ <
tube T T communicating with a horizontal tube A B, at the extr^B»t
mities of which A, B, are two apertures in opposite directions. Wh— ^
water from the millcourse MN is introduced into the tube TT, H
flows out of the apertures A, B, and by the reaction or count^^'
pressuro of the issuing water the arm AB, and consequently {9^e
whole machine, is put iu motion.
In order to understand how this motion is produced, let us svp*
pose both the apertures shut, and the tube T T filled with water up
to T. The apertures A and B, which ore shut up^ will be presseJ
outwards by a force equal to the weight of a column of water whose
height is T T, and whose area is the area of the apertures. Every
part of the tube A B sustains a similar pressure ; but as these pres
sures arc balanced by equal and opposite pressures, the arm A B is at
rest. By opcnhig the aperture at A, however, the pressure at that
place is removed, and consequently the arm is carried round bv a
pressure equal to that of a column TT, acting upon an area equal to
that of the apei'ture A. The same thing happens on the arm TB;
and these two pressures drive the arm A B round in the same direc
tion. This machine may evidently be applied to drive any kind of
tnachinery, by fixing a wheel upon the vertical axis CD.
8. Mr. Runisey, an American^ and Mr. Segner, improved this
machine, by conveying the water from the reservoir, not by a pipe,
in greater part of which the spindle tuklis, but by a pipe which de
scends from a reservoir, as F, until it reaches lower than tlie arms
A B, and then turns up by a curvilinear neck and collar, entering
between the arms at the lower part, as shown in fig. 229. This
greatly diminishes the friction.
CHAP. VII.] AIB AS A MKCHANICAL AGENT. 297
9. Professor Playfair has correctly remarked that the raoving force
becomes greater after the machine has hegan to move ; for the water
in the horisoDta) arms acquires a centrifugal force, hy which its pres
sure against the sides is increased. When the machine works to the
greatest advantage, the center of the perforations should move with
the TelociQr — '\/hg^ where r is the radius of the horizontal arm,
mearared from the axis of motion to the center of the perforation,
and r J the radius of the perpendicular tube, g being put for the force
of gravity, or 82 j^ feet.
As 2 V r is the circumference described by the center of each per
foration, — =r is the Ume of a revolution in seconds.
The quantity — y/hg is also the velocity of the effluent water;
therefore, when the machine is worked to the greatest advantaee, the
velodty with which water issues is equal to that with which it is
carried borisontallv in an opposite direction ; so that, on coming out,
it falls perpendicularly down.
10. The following dimensions have been successfully adopted;
vis. radius of the arms from the center of the pivot to the center of
the disdiarging holes, 46 inches; inside diameter of the arms,
3 inches ; diameter of the supplying pipe, 2 inches ; and height of
the working head of water, 21 feet above the point of discharge.
When the machine was not loaded, and had but one orifice open, it
made 115 turns in a minute. This agrees to a velocity of 46 feet in
a second for the orifice, being greater than the full velocity due to the
head of water by between 9 and 10 feet ; the difference is due to the
effect of the centrifugal force.
The the<»y of this machine is yet imperfect \ but there can be no
donbt of its utility in cases where the stream is small, with a con
stderable fidl.
Mr. James Whiteland, a correspondent of the Franklin Joumaiy
proposes to make the horizontal arms of the mill of a curved form,
BQch that the water will run from the center to the extremity of the
arms in a straight line when the machine is working. For the me
thod of constructing the curve, see Mechanics Magazine^ No. 499.
It is very clear, however, that the additional efficiency of the
machme will not be so great, by any means, as the inventor anti
cipates.
Sbct» IL Air a$ a Mechanical Agent
In the application of wind to mills, whatever varieties there may
W in thrir internal structure, there are certain rules and maxims vrith
298 AIR A8 A MECHANICAL AGENT. [PABT II.
regard to the position, shape, and magnitude of the sails, which will
bring them into the best state for the action of the wind, and the pro
duction of useful effect. These haye been considered much at laige
by Mr. Smeaton ; for this purpose he constructed a machine, of
which a particular description is given in the Pkilosopkieal Tranm»'
tions^ vol. 51. By means of a determinate weight it carried round an
axis with an horizontal arm, upon which were four small movemble sails.
Thus the sails met with a constant and equable blast of air ; and at
they moved round, a string with a weight affixed to it was wound
about their axis, and thus showed what kind of size or construction
of sails answered the purpose best. With this machine a great
number of experiments were made : the results of which were as fol
lows : —
(1.) The sails set at the angle with the axis proposed as the best
by M. Parent and others, viz. 55% was found to be the worst pro
portion of any that was tried.
(2.) When the angle of the sails with the axis was incrcaiied from
72° to 75°, the power vths augmented in the proportion of 31 to 45;
and this is the angle most commonly in use when the sails are
planes.
(3.) Were nothing more requisite than to cause the sails to acquire
a certain degree of velocity by the wind, the position recommended
by M. Parent would be the best. But if the sails are intended, with
given dimensions, to produce the greatest effect possible in a given
time, we must, if planes are made use of, confine onr angle within
the limits of 72° and 75°.
(4.) The variation of a degree or two, when the angle is near the
best, is but of little consequence.
(5.) When the wind falls upon concave sails it is an advantage to
the power of the whole, though each part separately taken tihonld not
be disposed of to the best advantage.
(6.) From several experiments on a large scale, Mr. Smeaton has
found the following angles to answer as well as any. The radius is
supposed to be divided into six parts ; and ^th, reckoning from the
center, is called 1, the extremity being denoted 6.
No. Angle with th??Si2^
1 72° IS''
2 71 19
3 72 18 middle
4 74 16
5 771 121
6 83 7 extremity.
(7.) Having thus obtained the best method o£ weat^erin^ the sails,
t. e. the most advantageous manner in which they can be placed, our
author's next care was to try what advantage could be deriTed from
an increase of surface upon the same radius. The resolt obtained
was, that a broader sail requires a larger angle ; and when the shI
(Nil proper lur ciiiiirgtfu hmia ; ii uciiig luuiiu in prauucc umi uiis
bould rather be too little than too much exposed to the direct
of the wind.
nc have imagined, that the larger the sail the greater would be
iwer of the windmill, and have therefore proposed to fill up the
area ; and by making each sail a sector of an ellipsis, acconling
Parent's method, to intercept tlic whole cylinder of wind, in
to produce the greatest effect possible. From our author's
iments, however, it appeared, that when the surface of all the
zeeedcd seveneighths of the area, the effect was rather dimi
I than augmented. Hence he concludes, that when the wliolo
er of wind is intercepted, it cannot then produce the greatest
for want of proper interstices to escape.
t is certainly desirable," says Mr. Smeaton, ^' that the sails of
nills should be as short as possible ; but it is equally desirable,
be quantity of cloth should be the least that may be, to avoid
>e by sudden squalls of wind. The best structure, therefore,
"ge mills, is that where the quantity of cloth is the greatest in a
circle that can be: on this condition, that the effect holds out
^portion to the quantity of cloth ; for otherwise the effect can
Rented in a given degree by a lesser increase of cloth upon a
radiua than would be required if the cloth was increased upon
me radius."
) The ratios between the velocities of windmill sails unloaded,
rhen loaded to their maximum, turned out very different in dif
; experiments; but the most common proportion was as 3 to 2.
aeral it happened that where the power was greatest, whether
enlargement of the surface of the sails, or an increased velocity
! wind, the second term of the ratio was diminished.
) The ratios between the least load that would stop the sails
le maximum with which they would turn, were confined be
that of 10 to 8 and 10 to 9 ; being at a medium about 10 to
Old 10 tn 0. or nhnnt A to /i; thnnrrh nn th<» whnio if orkrMMii*o«1
300 SUSATON's bulbs fob windmills. [pAHT 11.
mils in ten or fifteen minutes ; and, from the length of the arms from
tip to tip, has computed, that if an hoop of the same size was to ran
upon plain ground with an equal velocity, it would go upwards of
thirty miles in an hour.
(11.) The load at the maximum is nearly, but somewhat less than,
as the square of the velocity of the wind ; die shape and position of
the sails being the same.
(12.) The effects of the same sails at a maximum are nearly, but
somewhat less than, as the cubes of the velocity of the wind*
(13.) The load of the same sails at a maximum is nearly as the ^
squares, and the effect as the cubes of their number of turns in a^m
given time.
(14.) When sails are loaded so as to produce a maximum at m^
given velocity, and the velocity of the wind increases, the load con— <tf
tinuing the same; then the increase of effect, when the increase oft^s
the velocity of the wind is small, will be nearly as the sqnares oft^o
these velocities: but when the velocity of the wind is double, th^^:
effects will be nearly as 10 to 27; and when the velocities com— jC3
pared are more than double of that where the given load produces i^
maximum, the effects increase nearly in a simple ratio of the velocit)^'
of the wind. Hence our author concludes, that windmills, snch wm^
the different species for draining water, &c, lose much of their effecft^
by acting against one invariable opposition.
(15.) In sails of a similar figure and position, the number of tium^ ^
in a given time will be reciprocally as the radius or length of the^ ^m
sail.
(IG.) The load at a maximum that sails of a similar figure ■w i j
position will overcome, at a given distance from the center of motioo0^
will be as the cube of the radius.
(17.) The effects of sails of similar position and figure are as tb«
square of the radius. Hence augmenting the length of the sail with
out augmenting the quantity of cloth, does not increase the power;
because what is gained by length of the lever is lost by the slownen
of the motion. Hence also, if the sails are increased in length, the
breadth remaining the same the effect will be as the radios.
(18.) The velocity of the extremities of the Dutch sails, as well
us of the enlarged soils, either unloaded or even when loaded to a
maximum, is considerably greater than that of the wind itself. This
appears plainly from the observations of Mr. Fciguson, already re
lated, concerning the velocity of soils.
■ (19.) From many observations of the comparative effects of aaih
of various kinds, Mr. Smeaton concludes, that the enlarged sails are
superior to those of the Dutch construction.
(20.) He also makes several just remarks upon those windmills
which are acted upon by the direct impulse of the wind against aaila
fixed to a vertical shaft : his objections have, we beliere, been joa*
tified in every instance by the inferior efficacy of these horiaontal
mills.
^* The disadvantage of horisontal windmillsj" he remarks, ^ does
CHAP. VII.] coulomb's EXPBRIMBNTS. 301
not consist in this, tliat eacb sail, when directly opposed to the wind,
is capable of a less power than an oblique one of the same dimen
sions ; but that in an horizontal windmill little more than one sail
can be acting at once : whereas in the common windmill, all the four
ad together ; and therefore, supposing each Tane of an horizontol
windmill to be of the same sise with that of a vertical one, it is mani
fest that the power of a vertical mill ^nll be foar times as great as
that of an horizontal one, let the number of vanes be what they will.
This disadvantage arises from the nature of the thing ; but if we
oonsider the farther disadvantage that arises from the difficulty of
getting the sails back again against the wind, &c., we need not won
der if this kind of mill is in reality found to have not above one
eighth or onetenth of the power of the common sort ; as has ap
pMred in some attempts of this kind."
coulomb's experiments.
M. Conlomb, whose experiments have tended to the elucidation of
many parts of practical mechanics, devoted some time to the subject
of windmills. The results of his labours were published in the
Memoirs of the Paris Academy for 1781. The mills to which he
directed bis attention, were in the vicinity of Lille, and were, in fact,
oil mills. From the outer extremity of one sail to the corresponding
extremity of the opposite sail, was 70 feet, the breadth of each sau
•J feet, of which the sailcloth when extended occupies 5j feet,
bemg attached on one side to a very light plank ; the line of junc
tion of the plank and of the sailclod), forms, on the side struck by
the wind, an angle sensibly concave at the commencement of the
sail, but diminishes gradually all along so as to vanish at the remoter
extremity. The angle with the axis, at seven feet from the shafl, is
60*, and it increases continually so as to amount to nearly 84"^ at the
extremity. The shaft upon which the sails turn, is inclined to the
horizon, in different angles in different mills, from H'' to 15^.
Coulomb infers from his experiments,
(I.) That the raUo between the space described by the wind in a
second, and the number of turns of a sail in a minute, is nearly con
stant, whatever be the velocity of the wind ; the said ratio being
about 10 to 6, or 5 to 3.
(2.) That with a wind whose velocity is 21^ feet per second, the
quantity of action produced by the impulsion of the wind is equiva
lent to a weight of 1080 pounds avoirdupois raised 270 feet in a
minate; the useful effect being equivalent to a weight of 1080
pounds raised 232 feet in the same time : whence it results that the
quantity of effect absorbed by the stroke of the stampers, the fric
tion, &c., is neariy a sixth part of the quantity of action.
(3.) Suppose one of these mills to work eight hours in a day,
Coolomb regards its daily useful effect as equivalent to that of 1 1
horses working at a walking wheel, in a path of the usual radius.
(4.) It is obaemble, that in most windmills the velocity at the
802 THE STEAM ENGINE. [pART II.
extremity of the sails is greater than tbat of the ^iiid. In some
coses, indeed, these velocities have been found in about the ratio of
5 to 2. Now, it is evident that the impulsion of a fluid a^inst any
surface whatever, can only produce pressure, or mechanical effect,
when the velocity of the surface exposed to the impulse is less than
that of the fluid; and that the pressure will be nothing when the
velocity of the surface is equal to, or greater than that of, the fluid.
Indeed, in the latter case, the pressure may operate agaimt the
motion of the sails, and be injurious. It is desirable, therefore, in
order to derive from a windmill all the effect of which it is sus
ceptible, so to adjust the number of the turns that the velocity of the
extremity of the sails bhall be IcMy or, at most, equal to that of
the wind.
It would be highly expedient to make comparative experiments
on windmills, with a view to the determination of that velocity of
the extremity of the sails which corresponds with the maximum
of effect.
If v denote the velocity of the wind in feet per second, t tbe time
of one revolution of the sails, A the angle of inclination of tbe sails
to the axis, and D the distance from the shaft or axis of rotation
the point which is not at all acted on by the wind, or beyond whictx
the sailcloth ought to be folded up ; then theoretical considerations
supply the following theorem, viz. : — *
D = '1092 ft? tan A.
Ex. Suppose 9 = 30 feet per second, t =2*25 seconds, an</
A = 75°; then
D = 1092 X 30 X 225 x 373205 = 27509 feet
This result agrees nearly with one of Coulomb's experiments, in
which the velocity of the wind was 30 feet per second, the fiails
made 17 turns in a minute, and they were obliged to fold up more
than 6 feet from the extremity of each sail, which were 34 feet
long, to obtain a maximum of effect. The angle A at that distance
from the tip of the sail was 75° or 76°.
Sect. III. Mechanical Agents depending upon Heat: — the Steam
Engine,
In tbe steam engine the moving power is derived from tbe yapoiir
produced from water, by the action of heat. Vapour is formed from
water under ordinary circumstances at its usual temperature, wbat^
ever that may be; but the rapidity with which it is formed, and its
elastic force, vary with the temperature and the pressure to which
the water is subjected.
In order to a proper understanding of this subject, it will be
necessary to explain the exact meaning of a few of the terms em
iniiy. ine uensiiy or specinc gravuy or sieam, is, uiereiore,
Iv as its specific volume. Tlie pressure^ tensiouy or ekutic
f the steam, is the force expressed in pounds which it exerts
every square inch of the interior surface of any vessel con
it.
en water contained in an open vessel is exposed to any source
, its temperature becomes elevated, and a portion of the water
erted into vapour and passes otf in that form ; this elevation
les until the temperature (as indicated by the thermometer)
! 212% when the vapour is formed with such rapidity as to
e a considerable commotion in the water, which is then famili
id to boil. Notwithstanding, however, that fresh quantities of
« (as before) being poured into the water, no further elevation
emperature above 212^ can be produced so long as the vessel
8 open. But, if we now close the vessel, so as to prevent the
from escaping as it is formed, the sensible temperature of the
will again begin to rise, and will continue to do so as long as
It is applied ; it will also be found that the temperature of the
in the upper part of the vessel will always be the same as that
water : and further, that as its temperature increases, so will
*ic force, or the effort which it makes to escape, also increase ;
g found that steam contained in a closed vessel in contact
vater and exposed to any given temperature will, under all
stances, be of the same density and have the same pressure,
lowledge of the nature of heat, of the precise mode in which
"mtes in the production of steam from water, of the species of
lation between the heat and the particles of the water, and of
uige (if any) which the particles of the water undergo in their
sion into steam, is so imperfect, that we are not able, by any
JdouB or reasoning, to arrive at the law which subsists between
isible temperature and the pressure of steam in contact with
; as, however, the determination of the relation which they
■\ ^€uA\ tf\f)iAti ia Aoeonfial f/k orrivini* of nrsTrt^nt rAonUo in /\tii*
304
THB STEAM BNOINB.
series, are the following, which have been collected bj
Pambour.
Mil
III
Author.
Value of p = the preuuxe
In lbs. per square inch.
Value of f = the 1
bjrFahrenl
Below 1
atmofphere
From 1 to 4
From 4 to
50
Southern
De Pambour
{.SflSS.}
(III.)
/ 98806 h t y
\ 198MS /
(V.)
(•S69704 + '0060091 1)*
(11.)
15$72S6 (p  OiMi
(IV.)
198*062 ji* 
(VI.)
14««1 p* 
As these formula are of universal importance in all in
nected with steam, whether as applied directly to the bU
or to other purposes, such as warming buildings, &c., anc
peculiar form having fractional or very high powers and
comes necessary to employ logarithms in their applicatic
express them logarithmically ; and, for the convenience o
are not conversant with algebraical formulae, we shall
reduce the logarithmic formulss to verbal rules, which we
trate by a few examples.
Expressed then, logarithmically, Southern's formulae be
I. •.. Log(j» — 0494.8)=: 513 log(513 +0 — 11
II. ... Log (t f 513) = 21923601 + ^^^, ,^
5*lo
De Pambonr*s —
III. ... hogp = 6 log (98806 + — 137873772;
IV. ... LogC^ + 98806) = 22978962 f
logp
VI. ... Log (i + 39644) = 21672906 +
And those of Dulong and Arago —
V. ... LogjD = 5 log (269704 ± '0068031 0;
logp
Or reduced to verbal rules : —
I. To find the pressure by Southerns formula. — Add
temperature of the steam, multiply the logarithm of i
5*13, and from the product subtract 112468073, the ret
be the logarithm of a number, to which, if we add 049
will represent the pressure in pounds per square inch.
m.] TBS BTEAU EKQIVM. 305
npie. Required the pressure of steam at the temperature of
Log of (150 f 513 = 2013) = 23038438
Moltipljring bj 513
11818718694
SubtractiDg 11246807300
Logof 373174 = 57191 1394
•73174 — 04948 = 3 78122 = the pressure required.
fV Jind tke temperature hy SoutherriB formula,— From the
\ in pouods subtract *04948, and divide the logarithm of the
br bj 5ldy to the quotient add 2*1923601, and the sum will
ogarithm of a number, which, if we subtract 51*3 from it,
fesent the temperature of the steam.
1^. What is the temperature of steam whose pressure is
per square inch ?
Log of (10 — 04948 = 995052) = 9978458
Dividing bj 5*13
•1945118
Adding 21923601
Log of 24370 = 23868719
••. 24370 — 51*3 = 192*4, the required temperature.
To/wrf the pressure by De Pamhour 8 formula, — Add 98 806
temperature, and 6 times the logarithm of their sum, minus
3772, will be the logarithm of the required pressure.
^pk. What is the pressure of steam at 247° ?
Log of (247 + 98806 = 345806) = 25388325
Multiplying by 6
152329950
Subtracting. 13*7873772
Log of 27901 (pressure required) = 1*4456178
T^iind the temperature by De Pambour'i formula.— Divide
■ithm of. the pressure by 6, and to the quotient add
•t, the sum will be the logarithm of a number, which, if we
•8*806 from it, will equal the temperature of the steam.
806 THB 8TBAM SNQINB. [PIBT
Example. Required the temperature of steam having a pi
of 35 Ihs.
Log of 35 = 1*5440680
Dividing l>y... 6
•2573447
Adding 22978962
Log of 359 121 = 2 5552409
.. 359121 — 98806 = 260215, the required temperature.
V. To find the pressure hy Dulong and Arago*s formula.— H '^
dply the temperature hy 0068031, and to the product add 2697^
then 5 times the logarithm of their sum will he the logarithm of W
pressure.
Example. What is the pressure of steam having a temperature
330°?
Log of (0068031 X 330 + 269704 = 2514727) = •4004909i
Multiply by S
Log of 100566 (pressure required) = 2^02454g
VI. To find the temperature by DuiUmg and Aragos formula
Divide the logarithm of the pressure by 5, and to the quotient 6
21672906, and the sum will be the logarithm of a number whicbi
we subtract 39*644 from it, will equal the temperature.
Example. What is the temperature of steam at the pressure
120 lbs.?
Log of 120 == 20791812
Dividing by ... 5
•4158362
Adding 21672906
Log of 382937 = 25831268
.  . 382937 — 39*644 = 343293 = the temperature iequire<3
In any boiler, or other evaporating vessel, if the source of ft
be quite uniform, so that exactly equcJ portions of heat are giveim
to the water in equal intervals of time, it will be found that the t
of increase in the sensible temperature of the w%^r will dimihi^
as the sensible temperature itself increases; that is, that the Uaof
rature of the water will rise (for example) from 60'' to 70'' in A
heat thus required to convert any given weight of water into
diminishes as the temperature of the water increases, and is
that the sum of the sensible temperature and the latent heat is
I constant and equivalent to 1 170° of Fahrenheit's thermometer,
this circumstance two important consequences follow, viz., that
1 weight of steam, whatever may be its pressure and sensible
rature, will really contain exactly the same amount of heat,
ill therefore require precisely the same quantity of fuel for its
•sion from water at 212° to steam at the given density, what
hat may be ; or in other words, that the same weight of fuel is
ed to evaporate a given weight of water, and convert it into
whether it be contained in an open vessel, or closed and sub
to any pressure, and this result has been amply confirmed by
ment. The second consequence is, that if a quantity of steam
larated from the water with which it was in contact, and then
•d to expand into a larger space, or by pressure compressed into
ler, 80 long as it does not lose any portion of its own heat, or
e any fresh heat from surrounding bodies, its sensible tempera
ill always be precisely such as it would have been at the same
re when in contact with the water from which it had been
ced. And as in the steam engine, under ordinary circum
8, and where the usual precautionary means are adopted for
iting loss of heat by the steam in its passage to the cylinder,
mperature may without any practical error be supposed to suf
loss from those causes ; it results from the property which we
'xplained above, that the temperature of the steam in the cylin
the engine and its pressure will always bear the same constant
►n to each other that we have already shown they do while in con
ith the water in the boiler ; and, therefore, the formulee and
already given may be applied to determine either the tempera
r pressure of the steam (one being known) in any part of its
»B through the engine.
len water contained in a closed vessel is subjected to the con
I action of heat in the manner we have already supposed, the
ireture of the steam, together with that of the water, gradually
«e8y and fresh quantities of water are converted into steam;
rom both these causes that the elastic force or pressure of the
increases with the elevation of its sensible temperature. It
"esults, that as fresh quantities of water are converted into
, while the space which contains it is but very slightly increased,
X 2
308 THE STEAM ENGINE. [PART II.
the density of the steam must rapidly increase as the temperature
rises, and its specific volume being inversely as its density, must
become proportionally diminished. As the real quantity of water
contained in the steam which passes through the engine, (and which
depends upon the specific volume of the steam,) is one of the ne
cessary elements required in the calculation of the power of a steam
engine, it becomes of importance to investigate the changes which
steam undergoes in its density with any change in its temperature or
elastic force.
Let, therefore, V be the specific volume of steam at any tempera
ture t; then, since steam as well as all other aeriform fluids expand
^Q th part of their bulk at 32° for every degree that their sensible
temperature is raised, putting V ^ for the specific volume of steam
at 32°, we have
V^ (f32)
480
This formula supposes that no change has taken place in the
pressure of the steam during its change in temperature, but as we
have just shown that during all the changes which the steam under
goes in its passage through the engine, there is a constant relation
between the sensible temperature of the steam and its pressure ; it
therefore becomes necessary to determine in what way the specific
volume of the steam depends upon its pressure. Now, when steam
is separated from the water which produced it, and enclosed per se
in any vessel, it then follows the same law relative to its density as
any other aeriform fluid. This law has been already given at
page 279, and is as follows, viz. : — that wliile the temperature re
mains the same, the elasticity of the steam, or its pressure against the
sides of the vessel containing it, varies inversely as the space which
it is made to occupy, or, directly as its density.
Therefore, if P be put for the pressure of steam at 212% when its
specific volume equals V, and p^ be put for its pressure, supposing
its temperature unchanged but that its specific volume has become o,
then,
P : jt>j : : V : V
P
.. t?=V — .
Pi
If, however, t^ represent the temperature which steam of the
pressure p^ should have, and v^ represent its specific volume at that
temperature, we shall have
^ * 480 ^ ^ 480 '
then, substituting the value of v obtained above, and reducing in
respect of r^, we obtain
_ P 448jM,
'"''' p,' 660 •
CHAP. VII.] TUB STEAM ENOINS. 309
Now, it is found that V, or the specific volume of Bteam at 212% is
1700, that is, steam at 212° occupies just 1700 times the space that
the water from which it was formed occupied, and in this state its
pressure, or P, is equal to one atmosphere, that is, thirty inches of
mercury, or 14*706 lbs. on the square inch. Therefore, substituting
these values of V and P in the preceding formula, we obtain
16969 437879^, ^.,.,,
Pi = ' (VII.)
Pi
By this formula it appears to he necessary that both the tempera
tore and pressure of the steam should be known, in order to arrive
at its specific volume ; but we have already shown that the tempera
ture and pressure of the steam, both in the boiler and in its passage
through the engine, preserve an invariable relation to each other, so
that one can always be expressed in terms of the other; and we
can therefore, by combining this formula with those already given,
eliminate t^ altogether, and thereby obtain the value of e?. in terms
of /?j only. But although the three sets of formulae whicn we have
given, taken separately express this relation for a certain portion of
the scale of pressures, with a sufficient degree of accuracy for all
practical purposes, they will not admit of being correctly employed
beyond the limits assigned to each. As, however, in the investiga
tion of the changes which take place in the pressure of the steam in
the cylinder of the engine, it becomes essential to have some general
formula which shall express with equal exactness this relation be
tween the specific volume and the pressure, for all temperatures and
pressures wfiich can occur in the working of the engine, none of the
foregoing formulae are sufficiently comprehensive to serve for this
purpose; since, for example, the steam in an expansive engine may
pass from a pressure of five or six atmospheres to that of one and a half
to two atmospheres, which case would require the employment of
both formulee (VI.) and (IV.), in order to discover the corresponding
changes which had taken place in its specific volume.
In order to remove this objection, Navier has proposed the follow
ing general formula, which expresses the specific volume of steam,
in terms of its pressure only, and which, from the simplicity of its
form, is peculiarly well adapted for the purposes of calculation 4^ ; it is
v = ? (VIII.)
in which v is the specific volume of the steam, having the pressure
J} in lbs. per square foot; n and q are constants to be determined by
* The following formula, derived from an expression for the relation
between the pressure and temperature, given by the writer of the article on
the Steam Engine, in the Encyclopaidia Metropolitana^ gives the specific volume
of steam for all pressures above one atmosphere with singular exactness, but
its complicated form prevents its being applied in the subsequent investigation.
_ 1266424 80861 6
^  p "^ p •"*"** *
310 THE STEAM ENGINE. [PABT IT.
experiment, and the values of which, according to De Pamhour,
should he as follows, viz. : — For condensing engines,
n = 00004227, and q = 000000258 ;
and for noncondensing engines,
n = 0001421, and q = 00000023.
The reason of the distinction being made between condensing and
nonK:ondensing engines, is that the first values of n and q are found
more correct for low temperatures, and the second values for high
temperatures.
If we represent by S the space actually occupied by a given weight
of steam at the pressure /?, and whose relative volume is o, and by S j
the space which the same weight of steam will occupy at the pres
sure /7, and whose relative volume is v^, we have the following pro
portion : —
V : v^ :: S : Sj ;
and substituting for v and v. their values in terms of p and p^y bs
derived from equation (VIII.) we have
1 1
n + q p n + qp^ '
and reducing in respect of/?, we obtain
^=:'G +'.)? "^•>
Having thus explained the mechanical properties of steam, as far
as is requisite for developing the theory of the steam engine, we
shall conclude the subject by giving a table of the temperature and
specific volume of steam for pressures varying from 5 lbs. on the
square inch to 1 atmospheres ; the fourth column of which is cal
culated by FormulfiB (II.), (IV.) and (VI.), and the fifth colomn by
Formula (VII.)
CHAP. VII.]
THB STEAM BNGINB.
311
TABLE
OF THB PBB8SURB, TBMPBRATURB, AND SPECIFIC VOLUME OF
FBOM 5 TO 150 POUNDS ON THB SQUARE INCH.
STEAM,
II
II
ll
k
h
1
M
li
II
li
h
H
=1
III
iilii
T20
5
1614
4617
7056
49
2810
664
Mi
6
1692
3896
7200
50
282'3
563
1008
7
1760
3376
7344
51
2836
643
1152
8
1820
2983
7488
52
2848
684
1296
9
187'5
2673
7632
68
2860
625
1440
10
1925
2426
7776
64
2872
615
1584
11
1974
2221
7020
65
2884
607
1728
12
2013
2050
8064
66
2896
499
1872
13
2059
1905
8208
67
2007
490
2016
14
2001
1778
8462
68
2919
483
1
2160
15
2180
1669
8696
59
2930
476 i
2S0i
16
2164
1673
4
8640
60
2939
468
2448
17
219^6
1487
8784
61
2948
461
25^3
18
2226
1411
8928
62
2969
454
2736
19
2255
134S
9072
63
2970
448
2880
20
2283
1281
9216
64
2981
441
aos4
21
2310
1225
9360
65
2991
435
8168
22
2336
1173
9504
m
3001 1
429
3312
23
2360
1127
9643
67
301 '2
424
34 Se
24
2384
1083
9792
^8
3022
418
seoo
25 ;
2407
1043
9930
69
SOSiJ
412
3744
28
243*0
1010
10080
70
 3042
407
3888
27
2461
972
10224
71
3061
402
4m2
28
2472
940
10363
72
306'1
897
417e
29
2492
911
10612
78
3071
392
2
4320
30
2512
882
10656
74
3080
386
4484
31
2531
857
5
108(90
75
3089
382 1
4608
32
2550
832
10944
76
3099
878
4752
38
2568
809
11088
77
3108
373
4898
34
2586
787
11232
78
3117
369
5040
35
2603
767
11376
79
3126
366
£184
36
2620
747
11520
80
3136
361
dms
87
2637
729
11664
81
3144
357
6472
38
2653
711
11808
82
3152
363
5616
39
2669
694
11952
83
8161
349
5760
40
2684
678
12096
84
8169
345
5904
41
2699
663
12240
86
3178
841
6048
42
2714
649
12384
86
3186
338
6192
43
2720
636
12628
87
3194
334
6336
44
2743
622
12672
88
3203
331
3
6480
46
2767
609
12816
89
3211
327
6624
46
2771
697
6
12980
90
3219
324
6768
47
2784
635
13104
91
3227
321
6912
48
2797
574
18243
92
3236
318
812
THE 8TSAM ENGINE.
[part 11,
1
II
11
11
111
iHill
11
1
II
e
II
no
^1
hi
271
13392
93
3:24*3
315 1
15S40
3307
13536
H
325
312 1
16560
lis
3401
2m
130J>0
95
3258
3oa
@
17'iaO
120
343'S
250
13924
96
32fltf
30(J
ISOOO
125
3464
241
1396S
^7
3273
303
18720
130
3496
233
HI12
38
3281
mo
»
10440
135
35:24
225
lit25fi
m
3288
297
201S0
140
3553
217
144 im
urn
32P6
2i^S
20830
145
35S1
211
7
16120
1(15
3332
282
10
1
21600
150
360*8
204
GBNEBAL DESCRIPTION OF THE MODE OP ACTION OP THE STEAM
ENGINE.
Before proceeding to develope the general theory of the steam
engine, or to investigate the mode of action of the steam in the
cylinder, it will be well to enter into a general description of its
construction and mode of action. In this preliminary description we
shall only notice two forms of engine, namely, the non condensing
expansive engine, and the condensing engine without expansion.
Plate VII. exhibits a sectional elevation of a condensing steam
engine of eight horse power, with its boiler attached. The steam
being generated in the boiler A, is conveyed by the steampipe C
into the valve box or chamber I, which is shown upon a larger scale
in fig. 1, plate IX.; from this chamber there are three passages,
one (a) communicating with the top of the cylinder, the other (b)
with the bottom of the cylinder, and the third (c) with a yeasel to be
afterwards described, called the condenser. These passages are
covered by a sliding valve, </, so formed that when in the position
shown in fig. 1, all three of the passages are closed, but if it be
moved downwards, and brought into the position shown in fig. 2,
the upper passage a is made to communicate with the steam cham
ber I, while b and c are made to communicate with each other ; but
if it be moved upwards, and brought into the position shown in fig.
3, then the lower passage c communicates with I, and a and b with
each other. The action of this valve being understood, if we snppose
it to be in the position shown in fig. 2, we must immediately per
ceive that the steatn, which has been conveyed into the chamber I
by the pipe C, will pass by means of the passage a into the top of the
steam cylinder D, and by its pressure on the piston E give motion to
it, and cause it to descend. This piston is connected by a rod e with
a beam capable of turning about the center F, and the other end of
which is connected by the rod ^ with a crank hy upon the shaft of
which is fixed the flywheel O. It is therefore evident, ihftt any
CHAP. VII.] THE STEAM ENGINE. 313
motion of the piston £ will by means of the beam and connecting
rods be immediately communiatcd to the crank shaft h, and cause it
to revolve. Upon this shaft a contrivance termed an excentric (H)
is fixed ; this is nothing more than a circular disc, which has the shaft
or axis passing on one side of its center, the effect of which arrange
ment is to cause any point in its circumference to move nearer to or
further from its center as it is made to revolve, by which means a
reciprocating motion is given to the connecting rod k, which commu
nicates through the beam /, and rods m and n, to the slide valve d
already described. Now, when the piston E, by the continued pres
sure of the steam upon its upper surface, has been forced to the bot
tom of the cylinder D, this valve d will, by means of the excentric,
have been moved upwards, and brought into the position shown in
figure 1, in which the communication between the boiler and the
cylinder is closed, so that no more steam can be admitted to press
upon the piston £. The downward motion, however, of the piston
liaving been communicated, in the manner already described, to the
fly.wheel O, the momentum which it has thereby acquired causes it
to continue its motion, and as it carries round the crank ^, and the
excentric H will produce the following twofold effect ; first, it will,
through the instrumentality of the beam F, and connecting rods, e and^,
reverse the motion of the piston, and cause it to commence its ascent
in the cylinder; secondly, it will, by means of the excentric and
system of rods attached to it, cause the valve d to slide upwards, to
wards the position shown in fig. 3. As soon as this motion of the
slide valve commences, t)ie communication between the passage c and
the box I being opened, the steam will now pass to the bottom of the
cylinder, and there pressing on the lower side of the piston, will
cause its reascent.
We must now, however, notice what becomes of the steam which
already occupies the upper portion of the cylinder. By reference to
fig. 3, (which it must be recollected represents the position in
which the slide valve is now supposed to stand,) the upper part of
the cylinder will be seen to be in immediate communication, by
means of the passages a and b^ and the pipe O, with the condenser
M, which is a cylindrical vessel entirely surrounded with cold watet,
and which also has a small jet of cold water constantly playing into
iL If we now suppose the whole of the air to have been previously
expelled from the vessel M, (the manner of doing which will be pre
sently described,) so that nearly a perfect vacuum is formed within it,
it will readily be understood that the moment the communication is
opened between it and the upper part of the cylinder, the steam con
tained in the latter will rush into this vacuum with a very great velo
city, and being there immediately condensed Jby the cold water
playing into and also surrounding the vessel M, the vacuum will be
preserved, and thus no resistance will be offered to the motion of the
piston by the pressure of the steam upon its upper surface. As,
however, the injected water, together with that resulting from the
condeDBed steam, would in time fill the vessel M, an air pump, L, is
314 THE 8TBAM BNOINB. [PABT II.
made to communicate with it, which being worked by a rod from the
beam of the engine, always keeps the condenser empty. And fur
ther, as the heat lost by the steam would in a short time so far
elevate the temperature of the water surrounding the condenser as
to render it incapable of continuing properly to condense it, it be
comes necessary continually to change the water, with which object
a pump, P, worked by a rod from the engine beam, raises cold water
from a well or other source^ and pours a continued stream into the
vessel surrounding the condenser, while an equal quantity of the
warmer water is allowed to run off.
Before the engine is set to work, the cylinder D, the condenser
M, and the passages between them, are filled with common air,
which it is necessary to extract To effect this, by opening the
valves a communication is made between the steampipe C, the space
below the piston in the cylinder D, the eductionpipe O, and the
condenser M. The steam will not at first enter the cylinder D, or
will only enter it a little way, because it is resisted by the air ; but
the air in the eductionpipe O, and the condenser M, is forcibly
driven before it, and this part of the air makes its exit through the
valve N, called the suift valve, and which is kept covered with water.
The steamadmission valve is now closed, and the steam already
admitted is converted into water, in the manner already described, by
the coldness of the condenser M, and by the jet of cold water which
enters it from the well SS, in which the condenser is immersed.
When this steam is condensed, all the space it occupied would be a
vacuum, did not the air in the cylinder D expand, and fill all the
space that the original quantity of it filled ; but by the repetition of
the means for extracting a part of the air, the remainder is blown
out, and the cylinder becomes filled with steam alone.
In order that the connecting rod e may work freely, and yet
possess the desirable property of being steamtight, it passes through
what is called a stuffing or packing box. This stuffing consists of
some material which the steam will rather adapt to its office than
injure; leather, which answers well for the stuffing or collars of
machines never to be subjected to heat, will not answer here;
hempen yarn is the material usually employed. The rod of the
piston / passes through a stuffing box of the same kind as that of the
piston £ ; and the pistons themselves are surrounded with stuffing.
The cylinder D is surrounded by a case, to keep it from being
cooled by contact with the external air. The extremity, or any
given point removed from the center of the great beam, can describe
only the arc of a circle; but it is necessary that the piston rod e
should rise and fall vertically. An apparatus is therefore used,
called the parallel joint, which is easily understood by inspection.
By this means the rod e not only rises and falls perpendicularly, bat
is perfectly rigid, and communicates all its motion to the great beam
in each direction of its motion. The connectiivj rod g does not
require the same contrivance, because it does not rise and fall per
pendicularly; its lower end, with the outer end of the crank, describ
CHAP. VII.] THB STEAM ENGINE. 315
ing a circle : it has therefore only a simple joint, admitting of this
deviation.
In order to communicate a rotatory motion to the flywheel, in
stead of the crank may he used a contrivance giving twice the rapidity
to the fly. For this purpose, on the outside of the axis of the fly,
irhere the crank is shown in the plate, a small toothed wheel is
fixed, and can only he moved with the fly : at the extremity of the
rod g^ and on that side of it which is next the flywheel, another
toothed wheel is fixed, in such a manner that it cannot turn round
on its axis, hut must rise and fall with the rod to which it is attached.
These two wheels work in each other, and that attached to the connect
ing rod cannot leave its fellow, because their centers are connected by
a strap or bar of iron. When, therefore, the connecting rod rises, the
wheel upon it moves round the circumference of the wheel upon the
axis of the fly. By this means the fly makes an entire revolution for
every stroke of the piston, and some mechanics are apt to think that
they are great gainers by such an arrangement : the contrivance is
certainly el^ant, but with respect to utility, the fact is, that a crank
is preferable ; for it is more simple, cheaper, and less likely to be out
of order, while, if the fly be large enough to receive, with less velo
city, all the momentum that can be communicated to it, the efiect
will certainly not be inferior.
We now pass on to describe the noncondensing expansive engine
shown in plate VIII., in which fig. 1 is a side elevation ; fig. 2 an end
eleTation; fig.daplan; and fig. 4 plate IX., asection of the cylinder, show
ing the steam passages and valves. By a glance at this plate, the means
by which the motion of the piston is transmitted to the crank shaft
"mil be seen to be similar to that just described ; the real diflerence
between them consists in there being no condenser and air pump in
the engine which we are now describing, and in its having a second
ezcentric by which the admission of steam to the chamber I from the
boiler can be cut off at any desired portion of the stroke. The mode
of admitting the steam to the top and bottom of the cylinder, alter
nately, by means of the slide valve d^ is precisely similar to that
already described; but after the steam has done its duty in the
cylinder, instead of passing into the condenser it escapes by the
passage d, directly into the atmosphere. We have yet, however, to
explain the use of the second cxcentric, and to show in what manner
the steam is caused to expand in the cylinder, by which a saving is
eflTected in the quantity of steam used by the engine. In the engine
which we have just described, the steam is supposed to remain on
during nearly the whole stroke of the engine, and to have therefore
nearly the same pressure throughout the stroke ; but in that which
we are now describing, the communication between the boiler and the
cylinder can be closed at any period of the stroke that is desired, so
that after the steam has been cut off, it then expands in the cylinder
as the piston moves before it, and therefore presses upon the piston
at each instant with a pressure due to its specific volume at that
instant.
The method of adjusting the exccntric rod, so as to cut off the
316 THE STEAM BN6INE. [pABT II.
Steam from the cylinder at any desired period of the stroke, is shown
in fig. 5, plate IX., and is as follows: — ah is a bellcrank lever,
by means of which the motion of the excentric is communicated to
the valve, the arc cd being the distance (termed its effective stroke)
through which the valve has to be moved to cut off the steam entirely
from the cylinder; then, if we make the angle gkk equal to the
angle ckd^ and taking the diameter of the excentric's path, or double
the distance between the center of the excentric and the center of
the shaft in the compasses, apply them along the two dotted lines,
the point marked 6 where they fall upon both, or where the angular
distance is equal to the diameter of the excentric's path, will be the
f)oint at which the excentric rod must be fixed to the arm h of the
ever, in order that the steam may remain on during the entire
stroke. Then, fig. 6, representing the excentric's path, if it be
desired to cut off the steam at any portion of the stroke, as at ^, i,
or J, we have only to measure such a proportion of the semi
circumference of the excentric's path, and taking the length of its
chord in the compasses, apply them in the manner already described,
to the angle gkh^ and the distances from k^ at which they severally
fall upon both lines, will be the distances at which the excentric rod
should be fixed when it is desired to cut the steam off from the
cylinder at any such portions of the stroke.
Or, if we put / for the length of the arm a of the lever, f for the
effective stroke of the valve or the distance erf, ^ for the diameter of
the excentric's path, d for the distance from k to the point at which
the excentric rod should be attached to the arm h of the lever, and n
the fraction of the stroke at which it is desired to cut the steam off
from the cylinder, then
/^8in(90«)
E
For example, if / equal 6 inches, t equal 1*5 inch, and 1 equal
3 inches, and we give to n the successive values *25, '5, "75, or
suppose the steam to be cut off at J , \ and \ of the stroke, we shall
have for the corresponding values of rf,
, , C X 3 X 383 ^ ^ . ,
at J stroke d = z = **^ inches,
1*5
at ^ stroke d = = 8*5 „
1*5
1 . , , ^ 6 X 3 X 924
and at } stroke d = = 1 11 „
1*5
THKORV OF THE STEAM ENGINE.
Before entering uj)on the general theory of the steam engine, it is
but right to mention tliat the manner in which we purpose consider
ing the subject is tjimilnr to that first laid down by the Comte De
Pambour, in his very able work * upon this subject ; and at the
* The Theory of the Steam Engine, by Comte De Pambour. PubUalaed bj
John M'eale, 1839.
CHAP. VII.] THB 8TBAM ENGINE. 317
same time to acknowledge the great services which he has rendered
b? his investigations, which have led to the adoption of a theory
alike applicable to every form of engine, and working under all cir
cumstances. The propositions upon which his theory is founded,
are thus stated by him at page 25 of his work :^—
^^ From what has been stated, it plainly appears that we ground
all our theory on these two incontestable facts: 1st, that the engine
having attained uniform motion, there is necessarily equilibrium
between the power and the resistance; that is, between the pressure
of the steam in the cylinder^ and the resistance against the piston,
which furnishes the first relation *,
Pj = R.
And 2dly, that there is also a necessary equality between the
production of the steam and its expenditure, which furnishes the
second relation t,
_ w8 P
And these two equations suffice for the solution of all the
problems."
The manner in which Tredgold, and other writers upon this sub
ject, had determined the power of an engine, was by measuring the
area of its piston, and then, assuming that the engine would move at
a given velocity, and that the pressure in the cylinder would be
identical with that in the boiler, they thought that the continual
product of these quantities would give them the load which the
engine ought to raise in a given time, and to a given height, pro
Tiding no loss had arisen from the friction of the machine, and other
canses; and further, supposing that the portion thus lost always
bore the same constant ratio to the whole, in the same cla.ss of
engines, they determined arbitrary coefficients, by which they multi
plied the above product, and they conceived that the quantity thus
obtained represented the actual effective power of the engine with
sufficient accuracy for all praqtical purposes, although no notice
ivhaterer was taken in the calculation of the evaporative power of
the boiler.
A little consideration will, however, show the inaccuracy of this
method of calculation. The engine itself is not the moving power
by which we produce the desired effects; the boiler is the part in
^vhich the power resides, and the engine is only the instrument or
machine through which this power is transmitted, and by which it
can be applied precisely at the point, in the mode, and with the
* In whidi p, represents the pressure of the steam against the piston, per
unit of surfaoe, and a represents the resistance of the load against the piston,
divided in like manner per unit of surface.
•f In which v is the velocity oi the piston under the resistance a, m the
specific Tohime of the steam in the hoiler under the pressure p, or the ratio of
Its Tohime to that of the water which produced ii^ s the volume of water
er apo rafd by the boiler in a unit of time, and a the area of the cylinder.
318 THE STEAM BNOINB. [paR*^
velocity which we desire. A steam engine may, in this respect^
compared with a crane, by means of which a number of met^
enabled to raise a considerable weight ; bat here no power resid.ef
the crane, it is merely the instrument through which the mcT^ i
enabled to apply simultaneously their united energies, and to pro^i
certain effects; but those men could produce an equal amon^int
effect without the crane, although not with so much ease or con ^ei
ence. And, in like manner, the power which can be transmitt^
the engine can only equal that which resides in the boiler. For,
in the crane, the power exerted through it must depend on tl
number of men by whom it is moved, so also in the steam engfiK
must the power which it can exert, or the amount of work which i
can perform, depend only upon the power of the boiler. iVot
however, that we mean to assert, that any amount of power which
it is possible to derive from a boiler could be transmitted and app/fecf
by the engine, for, as the crane has been designed and proportfooed
only for the production of a certain effect, and the application of a
greater power than that required for its production might strain and
injure the machine, so it is with the steam engine, every engine is
constructed only to transmit safely a certain amount of power frotn
the boiler to the working point, and the attempt to produce a grease
effect might be attended by the derangement of the machinery.
If a equal the area of the piston in square feet, and p^ ^
pressure of the steam in the cylinder, per square foot of surface, tl^
product, or a/?, will obviously represent the resistance which ^
engine is capable of overcoming; and if this be multiplied by ^
length of the piston's stroke = /j , we shall have the whole amo^
of work which the engine will perform in each stroke, or represent^
this amount by k'i , we shall have
», = a/,/?, (X.)
This equation, however, supposes the steam to act with C
pressure pi during the whole stroke of the engine, and therefore on
applies to engines in which the steam is not cut off until the end •
the stroke, and which therefore do not work expansively.
In order then to obtain a general expression which will indac
the expansive engine also, let us suppose that /j represents only th
portion of the stroke during which the steam is not cut off, in whi<
case the above formula will still represent the work done by tl
engine during such portion of the stroke. Now, if x equals tl
length of a portion of the cylinder, equal in content to the stei
passages and the space left at each end of the cylinder for the clei
ance of the piston, we shall have a (/, + >^) for die actual space occ
pied by the steam in the cylinder (having a pressure of />, ) at t
moment when the steam is cut off; then, if/ represent the length of t
stroke from its commencement, at any moment after the expansi
of the steam has commenced, and p the pressure of the steam d
to its altered volume, we shall have from formula (IX.)
/, + X /w \ n
CHAP. VII.] THB STEAM ENGINE. 319
Then, since the work done by the steam in the cylinder is equal
to the pressare upon the piston multiplied by its area and the length
of the stroke, or the distance that the piston has moved under that
pressare, if A / represent the distance moved over by the piston after
the steam has been cut ofif^ we shall have for the work performed
daring such motion,
a/7 a/,
in which, substituting for p its value derived above, we have
therefore, the whole amount of work done by the steam during its
expansion in the cylinder will be represented by
or putting w, for this amount of work, and /, for the total length of
the stroke from its commencement,
«nd int^rating between the limits of /, and /,, we have
». = a (/. + X) hyper. Wj^ (^ + ^ ») " f « (^'  ^•)  (^I)
Now, if we add together the work done before expansion, (= t0„)
and the work done after expansion, ( = «;„) and represent the whole
work performed by the engine each stroke by W, we shall have
W = w, + w,;
in which, substituting for w^ and w, their values derived from equa
tions (X.) and (XI.), and reducing, we obtain
W=a Q + ;».) [K + (/. + A)hyper. log \±^^  ^ a /.....(XI/)
Then, if P represent the pressure or resistance which the work
W exerts against every square foot of the surface of the piston,
W = Pa^, and therefore
P/. = d + />.) {/. + (/. + X) hyper. logjf^} ]ir  (XII.)
wbich ezpreanon becomes
P=J», (XIII.)
when l^=zli^ or when there is no expansion.
These forrouls, then, enable us to determine the effects which
would be produced by any engine working under given circumstances;
hat it b^mes a matter of considerable practical importance to
320 THE STEAM BN6INB. [PART II.
determine those values of the several quantities which shall produce
the greatest mechanical effect with the least expenditure of steam ;
or, in other words, the quantity of water evaporated by the boiler
remaining constant, shall give to P its maximum value. In the high
pressure engine, working without expansion, this takes place when
the pressure in the cylinder is as nearly as possible equal to that in
the boiler. In the expansive engine, the same law holds with regard
to the pressure at which the steam should be admitted to the
cylinder to produce the greatest mechanical effect with any given
expansion. But there is in every engine working by expansion a
certain proportion of the stroke through which the expansion will
produce a greater effect than through any other proportion, and this
will be easily found after we have obtained expressions for Q, or the
number of cubic feet of water which the boiler is capable of evapo
rating per minute.
If V. equals the specific volume of the steam at the pressure /?, ,
at which it is admitted to the cylinder, we shall have for the num
ber of cubic feet of steam of that pressure, generated per minute,
Q f 1 ; and if the engine makes » strokes per minute, each equal
/j feet in length, »a(/j + ^) will equal the number of cubic feet
of steam at the pressure p^ used by the engine per minute, and
therefore from the second principle laid doivn by De Pambour*,
we have
Qr, = ar(/, H >);
then, if V equal the velocity of the piston in feet per minute,
V
we have V = i/^ , or » =  ; and also from equation (VIII.) we
have V, = ; therefore, substitutine these values above,
we obtain
Q aV(/, +x)
n + qp^ I,
whence we obtain for the value ofp j ,
_ l,Q n
(XIV.)
^* ayV(/, +A) g
We have also seen by equation (XIII.) that in an unexpansive
engine /^j =: p, p being the resistance against each square foot of
tlie piston's surface, arising from its load, from the friction of the
engine, and from the pressure of the atmosphere, or imperfectly
condensed steam on the other side of the piston. Now, taking the
most simple case, or that of the stationary engine, if we put ^ for the
resistance occasioned by the load upon each square foot of the
piston's surface, J* ^^'^ ^^^ resistance arising from the friction of the
engine when unloaded, and ^ ^ for the additional friction produced
* See page 317.
CHAP. YII.] THB STB AM BNOINB. 321
by the load ^ upon eyery square foot of the piston s surface, and also
r to represent die pressure of the atmosphere or uncondensed steam
upon each square foot, we have
and substituting this value of jp^ in equation (XIV.) above,
";*»>+/^'=.?vTi7T^? <^^>
wbich equation expresses the relation between all the several quan
tities for unesfpansive eneines.
In exfMUuive engines 7^ represents only that portion of the stroke
which IS performed before the steam is cut off, the whole stroke
being equal to /^ ; therefore in this case ' = j > cuid equation
(XIV.) becomes
'.=j?4rnr)F <''^''
Now, from equation (XII.) we have for expansive engines,
/, +(/,+x) hyper, log ^^ ^^
•nd substituting for P its value determined above, reducing, and
solving in respect of ^, we have
Q(rV, + typer.log^j) /+. + ^
( = ^1 ^ ^i, I M+^^  _ 1 ... (XVII.)
Then, aboe the actual useful mechanical effect produced by the
engine per minute equals ^ a V, we have for the power of the engine
^ ^(rTi + ''yp'^°grTD «v(/H^4f)
«"^ — TUTf^ — 1 + *
Now, in order to arrive at that value of /^ which will give the
maadmom value to ^ aV, or, in other words, to ascertain at what
iortion of the stroke the steam should be cut off in order to produce
the greatest effect, let us substitute for a V its value derived from
eqvfttion (XIV.), whence by reduction we obtain
9(1 +f)V/, + X^ '*^ */, +x l^+?, n+qp^ J
322 THE STEAM ENGINE. [PART II.
then, differentiating in respect of /^ and remembering that since the
above expression is a maximum its differential will = 0, we have
whence, reducing and solving in respect of j , we obtain
(XVIII.)
equal that portion of the stroke (the whole stroke being unity) at
which the steam should be cut off to produce the greatest effect, with
the least expenditure of steam.
DESCRIPTION OP THE VARIOUS KINDS OP ENGINES, AND THE FORMULAS
POR CALCULATING THEIR POWER.
Although almost every manufacturer of steam engines adopts his
own peculiar form of construction, by which a great diversity in
their external appearance is occasioned, the principles upon which
they act are similar in all engines of the same class; and, therefore, in
attempting a classification of the different kinds of engines, we shall
pay no attention to the details of their construction, but only regard
the mode of employing the steam in the cylinders for the purpose of
producing any particular species of motion. Proceeding, then, upon
these principles, the following table represents at one view the
various forms under which steam is at present employed as a
moving power.
Engines working  Noncondensing [ f^^'^,
«m^/ expansion  Condensing. ^
Engines working
with expansion
r"" [c^^ j^«
^ J Steam pressing only on the
c;,i»io i.^:»» J t upper surface of the piston.
V Single acting j  st^^„,^i„^ ^^U on the
I Steam pressing only on the
lower surface of the piston.
Stationary noneandenBtng EngineB without expansion.
The mode of action of the steam in the cylinders of this class < ^ "
engines is precisely similar to that already described at page 31^^
and a view of which is given in Plate VIII., with the excepUon tbi^^
the steam is never cut off from the valvebox I, and therefore pr ess e d
with an equal pressure upon the piston daring nearly the who^ff
extent of the stroke. With regard to the mechanical coDatmction a^
this kind of engine, it depends in a great measure upon the fancy c^
the maker, or upon the purposes to which the engine is to be applied ^
there is perhaps a greater diversity in the form of this class (^'«
engine than of any other. For particular examples of the method
CHAP. VII.] THE STEAM ENGINE. 323
of coDStnictiDg all kinds of CDgines, we must refer to the last edition
of Tredgold's elaborate work upon the steam engine, which, with
its appendices, forms one of the most magnificent works ever
published.
The formulse for calculating the power of engines of this class
may be immediately derived from equation (XV.) already given, by
solving in respect of any of the quantities which are wanted. Thus,
we obtain for the values of V^ Q, and ^,
v= ^^.
or, multiplying ^ by Vo, which then gives the actual work performed
by the engine in each minute, we have
We may reduce these formulae to a more practical form by insert
ing the numerical values of those quantities which are either constant,
or sufficiently so for all practical purposes. Thus, t being in this
case the pressure in pounds of the atmosphere upon each square foot
of the piston's surface, equals 2118 lbs. ; for n and ^, we have the
values already given for noncondensing engines at page 310, and
assuming that x = *05 /„, ^ = '14, and y = 144, as adopted by
De Pambour, the above formulae become
^ "■ a(6'95536 + 0027531 f)'
Vg (695536 f '0027531 e)
^ ■" 10000
^aV = 3632268 Q — 252637 a V.
Locomotive Engine working without expansion.
The mode of action of the steam in the cylinder of a locomotive
engine, does not in any way differ from that in the engine already
described; the only essential differences of which we have to take
notice being in the details of its construction, and in the nature of the
work which it has to perform.
Plate X. represents the longitudinal, and Plate XI. two transverse
sections of one of the engines employed on the Great Western Rail
way. The principal peculiarity in the construction of a locomotive
consists in the form of the boiler, which, in order to lessen the weight
fo be moTed, is much smaller than the ordinary boiler for other
Y 2
324 THB 8TBAM BNOINB. [PABT II.
engines of the same power, and wbicb, therefore, requires a very
intense heat in the furnace, and a large evaporating surface: the
first of these is effected by emp]o3ring coke instead of coal, and pro
ducing rapid combustion by a strong draught; and the second bv
causing the water entirely to surround the firebox, or furnace, and
by carrying the smoke and heated air from the same through a
number of small tubes which are likewise surrounded witli water.
Thus, it will be seen by an inspection of Plate X., and also of fig. 2,
Plate XL, (which latter is a transverse section through tlie firebox,)
that the water not only covers its upper surface, but that it surrounds
it on every side, leaving only a small space on one side for the
opening, I, by which the stoker regulates the fire. The tubes aa^
to which we have already alluded, for convejang the smoke from the
fire to the chimney, are ninetyone in number, and expose a very
large heating surface to the water ; and in addition there is a hollow
bridge, £, also filled with water passing through the center of the
fire. The bars, X X, of the furnace, are so arranged, that when it
is desired to put out the fire they can all be simultaneously lowered,
allowing the burning contents of the furnace to fall upon the ground,
or into a vessel placed to receive them underneath the engine. The
sides of the firebox, being flat, would be liable to be forced out by
the pressure of the water and steam, to obviate which the sides are
held together by short rivets, and the top is strengthened by a number
of short girders ^, bolted to its upper side. The flat ends of the
boiler are prevented from being forced out by the fire tabes aay
which are secured at each end, and by tie rods which run from one
end to the other; the other parts of the boiler derive snflicient
strength from its cylindrical form. There are two safetyvalves, one d,
under the immediate inspection of the enginedriver, and which by
means of a spring manometer Q, always indicates to him the pressure
existing in the boiler ; the other L, is placed (as an additional pre
caution) beyond his reach, so that it cannot be meddled with, or have
its pressure altered. A glass tube W, fig. 1, Plate XL, made to
communicate with the boiler at both its upper and lower extremities,
so that the water in it always stands at the same level as that in the
boiler, serves to indicate to the enginedriver when the quantity of
water falls short. An opening R, called a manhole, is left in the
upper part of the boiler, being covered with a screwed lid, in order
to enable the interior of the boiler to be examined ; c is a whistle,
so contrived, that the steam being allowed to rush forcibly through it— 
produces a very shrill whistle, which is employed as a very generaL^
and useful means of signaling on railways. It will be obsenred that^
the boiler is surrounded by an exterior case, a small space being lefk:^
between and filled with air, which prevents in a considerable degress
the loss of heat from radiation, that would otherwise take place.
In consequence of the steam space in a locomotive boiler beings
very limited, the steam is formed almost as it is used, and immedi'
atefy the communication with the cylinders is opened and the steaitt
allowed to flow into them, a violent commotion in the water in thff
CHAP. VII.] THE STEAlf ENGINE. 325
.boiler is occasioned by tbe sudden production of steam to supply its
place, the result of which is, that a large quantity of water, in a state
of minute mechanical division, is carried over with the steam into the
cvlinders, and there gradually accumulating, would at length materi
ally interfere with the working of the engine. This effect, which is
called primingy is to a certain extent obviated by a dome and steam
chest (H) heing formed at the top of the boiler, and by the entrance
of the steam pipe G by which the steam is conveyed to the cylinders
being carried to the upper part of this chest, and having its mouth
turned upwards, by which the space for steam is somewhat increased,
and more time is given for the separation of the water from the
steam. The admission of steam to the cylinders is regulated by the
Talve F, which the enginedriver can open or close by means of the
handle d. The steam pipe has an expansive joint Z, which allows of
a slight elongation or contraction of the pipe, in consequence of its
varvms temperature. The steam pipe passes straight through the
boiler mto the chamber in the fore part of the locomotive, termed the
smokebox, it there divides as shown in fig. 1, Plate XL, into two
hrancbesX/y ^ch leading to one of the cylinders; the manner in
which the admission of the steam alternately to the top and bottom
of the cylinder is effected by the slide valve T, is precisely similar to
that alr^y described at page 312. But in a locomotive it is neces
sary that Uie enginedriver should possess the means of making the
engine revolve in either direction, in order to effect which there are
two excentrics to each slide valve, fixed upon the crank shaft, oppo
site to each other, and so arranged that, by means of a lever within
reach of the enginedriver, he can connect either excentric with the
slide valve, and by reversing its motion also reverse that of the
engine itself: only one of these excentrics (C) is seen in the plate.
The motion of the piston n, is transmitted to the crank »«, by the
piston rod o, and connecting rod r; the piston rod being made to
move straight by the cross head B, moving in guides. After the
steam has performed its office in the cylinders it is expelled through
the pipe M, into the lower part of the chimney, where by its partial
condensation it produces a powerful current of air through the fire
and tubes aa. The water formed in the cylinder by condensation,
or that which is brought over by priming, is allowed to escape by the
cock P. The supply of water to the boiler is maintained by a small
forcepump D, the plunger of which is attached to the cross head B
of the piston rod, and which draws tbe water from the tender
through the pipe O, and forces it into the boiler. The cylinders,
bein^ entirely within the smokebox, are always maintained at a
considerable temperature, by which means very little loss results from
condensation of steam in the steam passages or cylinders.
The nature of the resistances to which a locomotive engine is
subjected are somewhat different to those of a stationary engine,
arising from the two following causes, viz., the resistance occa
sioned against the piston, by the ejected steam being blo\^ii forcibly
throDgb tbe contracted pipe M into the chimney, for the purpose of
326 THB STEAM ENGINE. [PART II.
producing a draught in the fire, as already described ; and the resistance
which the air occasions to the progressive motion of the train. The
former of these, the Comte de Pambour, by his experiments, has
ascertained to increase directly as the velocity of the piston, and to
amount upon an average to about 1*75 lbs. per square inch of the
piston's surface, when moving with a velocity of 150 feet per minute,
or putting tt ^ V for this pressure per square foot, at the velocity V,
we have
^1 : 175 X 144 : : V : 150;
.. «j = 16848.
The second resistance, both theory and experiments show to in
crease as the square of the velocity, and from the results of De
Pambour's investigations it appears that for a train of average length,
moving at a velocity of ten miles per hour, this resistance amounts to
about 33 lbs., which, assuming the engine to be of the most usual
dimensions, is equal to '0055136 lbs. for every square foot of the
piston, when moving with a velocity of one foot per minute; or,
putting r for this resistance, we have rV^ when the velocity is V.
As the value of r depends not only on the number and description
of the carriages composing the train, but also on the size of the
driving wheels, the diameter of the piston, the length of its stroke,
the gauge of the railway, and even on the direction and force of the
wind, it will be perceived that the value which we have assumed
above is very general indeed, and that, wherever great accuracy is
required, these circumstances must be taken into consideration, and
the more exact value of r deduced therefrom.
We have therefore, in this case, for the value of P, or the sum of
the resistances against the piston.
V =
m(X
of V,
l^
a(l, + ^){n + g[{^ + rV^)(l + (?) +/+ tt + », V]} '
which being substituted in equation (XV.) for ^ (1 + ^) f /"+ «, aDd_^
reducing, we have for the values of V, Q, and ^a Y, as follows : —
Or, substituting for n, ^, x, v and ^, the same numerical values i^b^
before, and making /*= 3*0125 lbs. per square foot of the piston'^ '
surface, when moving with a velocity of one foot per minute, whic^^
would be its average value, these formulae become,
10000 Q
^ "■ a (66143 + 002753^ + 00001518 7* + 00406887)'
CHAP. VII.] TH£ STEAM ENGINE. 327
Q = fQQ55(C*61*3 + 002753 e + 00001518 V« + 0040688 V);
^aV = 3632400 Q — a (005514 7^ f 1 4779 72 + 240249 V).
Condensing Engines working withotU expansion.
One fonn of this class of engine is shown in Plate VII., and its
general principles and mode of action have already heen descrihed
at page 312. This class includes the greater numher of engines
employed for propelling steamhoats, which, although different in
form, are identical in principle with the' engine shown in Plate VII.
The general formulae for calculating the values of V, Q and f a V,
are precisely the same as those already given at page 323, for non
condensing engines, and therefore need not he recapitulated here;
but the practical formulse differ in consequence of the coefficients
and constants having different numerical values. The value of 7,
or the resistance on the underside of the piston, is in this case much
smaller than in the preceding, and may be taken on the average as
only equal to 576 lbs. on each square foot ; the value of n and q
for condensing engines will be found at page 310, and those of x,
^, and fy may be taken the same as tliose already given at paee 323,
for stationary noncondensing engines. Substituting these values in
the general formulae, and reducing, we obtain
_ lOOOOQ
^ "■ a (2 3943 + 0030883 e) '
Y g (23943 f 6030883 {)
10000 '
^ a V = 3237280 Q — 77528 1 a V.
Stationary noncondensing Engines^ working ea!pansivefy. .
This engine, which is shown in Plates VIII. and IX., has been
already described at page 315, and it therefore only remains to
deduce the formulae by which the values of V, Q and faV may
be obtained.
Now we have, at page 321,
and from formula (Xyil.)> by reduction, we immediately obtain
828
THB 8TKAM BNOINB.
ygfnf y[g(l+<P)hj
[part II.
^ + hyper, log ±±
A
And subsUtuting in these formulae the nnmerical values of the
constant quantities, viz., w equal the pressure of the atmosphere,
equal 2118 lbs., n and g as given for noncondensing engines, at
page 810, and x and ^ as before; also, eiving to/* the value assigned
to it by Comte de Pambour for this class of engine, viz., 864 lbs.
per square foot of the piston's surface, we have
eav = 3813883 QJr4 + l»yP«' log r'"!  315773 a V;
10004 Q
V =
aC82796 + 002622 f)
yg (82796 f 002622 e)
In order to simplify the calculation of these Quantities, and ^^r
the benefit of those who are not conversant with the use of !<»£
arithms, we subjoin a table of the values of the expression
/.
/i + x
»i r A
hyper, log j , for values of j, (or the fraction ezpressmg tYjat
portion of the stroke which is performed before the steam is cut o^j
varying from onetenth to ninetenths of the whole stroke.
Value
Value of
Value
Value of
t
^ + hyper.log'« + ^
't
•10
261258
•60
166558
•15
240823
•65
1 •47628
•20
223507
•60
1^40366
•26
208610
•66
LS1218
•80
195576
•70
1^26981
•33
187721
•76
120943
•85
1 •84008
•80
116248
•40
178619
•85
109862
•45
164194
•90
104TS9
Loeonwtive Engine working expaiuively.
Lately the principle of working locomotive engines ezpannic
has been successfully adopted, and we therefore give the fumf
CHAP. VII.] THB STB AM BNOINB. 329
for calcolating the effects which thej will then produce.
In this case we have
*"'^ 7UT7)
^'«rv'.
And snhstitnting for the constant quantities the same numerical
ralues as those already employed at page 326, for locomotires work
ing without expansion, we have
^ IOOOOQ(.Ji_.^.Hhyper.log^^)
a (62993 f 002622 ^ + 00001446 V' + 003875 y) '
V a(62993 + 002622 e + '00001446 Y* + 003875 v)
10000(^^^ + hyper.log^^) '
^a V = 3813882 Q ( jA_ + hyper. log ^^^ )
— a (005514 y» + 14779 y« + 240249 V).
The apparent complication of these formula will disappear when
we snhstitute for f j— ^ — + hyper, log / ) its value from the
\ *i + ^ *i + ^ /
sbove tahle, which cannot he done until the proportion of the stroke
at which the steam is cut off is known.
StaHanaiy Condensing Engines^ with one Cylinder working
eapaneively.
This enffine only differs from that already described at page 312,
in its wonung expansively, which is effected by the agency of a
second excentric, in the manner explained at page 315.
The general formula for the values of V, Q, and e^^i ^^^ identical
with those already given at pages 327 and 328, for noncondensing
working expansively; but the practioJ formuln differ in
330 THE STBikM ENGINS. [PjkRT II.
consequence of the constants t, w, y, ?., ^, and f^ having in this case
the value assigned to them at page 327, for condensing engines working
without expansion ; these heing substituted in the general formulee,
they become as follows : —
10000 ^ /"^  i^™ i.«'« + ^
Q =
a (20953 + 0029412^)
Va (2*0953 f '0029412 6)
V = —
a (20953 + 0029412^)
^a V = 3399972 Q (j^ i hyper, log 2^) — 712123aV.
Condensing Engines, with two cylinders^ working expansively.
This form of engine was invented by Mr. Arthur Woolfe, and
patented by him in the year 1804. The peculiaritv in its mode of
action consists in employing two cylinders, through both of which
the same steam is made successively to pass. The details of its con
struction will be better understood by reference to Plate XII., in
which fig. 1 is a section of the two cylinders and slide valves.
Supposing the various parts to be in the positions shown in this
figure, the steam from the boiler enters by the aperture A, and
passes through the passage B into the top of the smaller cylinder
C, and there, pressing on the upper surface of the piston, pro
duces a downward motion; when the piston has performed a
certain portion of its stroke the communication with the boiler is
closed, so that the steam acts upon the small piston only by its
expansion during the remaining portion of its stroke. The rod a
of the small piston, and that b of the large one, being both connected
to the same beam, descend together, during which the steam which
occupied the space below the small piston passes through the
passages D and E into the space above the larger piston ; but, since
the content of the cylinder F is larger than that of C, in doing so it
expands and produces a pressure on the upper surface of the large
piston, the underside of which is in immediate communication with
the condenser by means of the passage O and aperture H. As the
pistons descend, the slide valves, moved by an ezcentric in the
ordinary manner, change their position, and during the upstroke of
the pistons are in the situation shown in fig. 2, in which it will be
seen that the steam from the boiler now presses directly on the
upper side of the smaller piston, while the top of the small cylinder
now communicates with the bottom of the larger one, by which the
effort of the steam, in expanding from one cylinder to the other, is
exerted on Uic under surface of the large piston, the upper side of
the same being now in direct communication with the condenser by
means of the passage £ and aperture I. The mode of action of the
CHAP. VII.] THE STEAU ENGINE. 331
steam in the cylinders being then correctly understood, it is only
necessary to state, that the manner of transmitting the motion of the
pistons to the working point, and the arrangement of the condenser
and air pump, may be the same as in other steam engines, and
similar to that which we haye already described.
In order to deduce formulee for calculating the effects of this class
of engine, let a^ represent the area of the small piston, and a^ that
of the larger piston, both in square feet; let /, equal that portion of
the small piston's stroke which is performed before the steam is cut
off, l^ the whole length of the small piston's stroke, and l^ the
length of the larger piston's stroke ; also, let p ^ be the pressure of
the steam in the small cylinder before the expansion commences,
Pa the pressure of the same in the large cylinder before expansion,
and ^, and A^ the length of a portion of each cylinder, equivalent
in content to the space left for clearance and the steam passages.
Then we have from equation (XI.*), page 319, for the work down
in the small cylinder,
«i (^ rp, ) 1^ f (/x + Xi) hyper, log ^^^^  ^«i U
Now, in order to obtain the work done by the expansion of the
steam in the large cylinder, put «, = a, /^ = the space moved
through by the small piston, «, =0^/3 = the space moved through
by the large piston, and « ^ a, / = Vie space moved through by the
same at any portion (/) of its stroke. Then, the space occupied by
the steam (having a pressure equal to p^) before the commencement
of the stroke, equals 9^ f^i^i f^^Xg, and that which it occupies
at any portion (/} of the strolce equals «fa,X {• a^x^; therefore,
patting 3 = ajAj Ha^Ag, we have from formula (IX.), for the
preasare of the steam due to its altered volume,
n \
Then, since A« represents the elementary space moved through by
the larger piston, we have for the work developed during the motion,
pAty which, by substituting the above value of/?, becomes
^. As /n \ n
therefore, the whole amount of work developed is represented by
('.+^)J:i^( +/»,} J:a.;
which being taken between the proper limits of « = « , and « = «j '
equals
<"*«/:'.Ti(=+'.)5/:^
332 THE 8TBAM BNOINB. [PABT II.
whence, by integnting, we obtain
(,, +^)hyper.log^l± (J +/>t )  ^(*, "'i)
Then, in order to eliminate />,, we can derive its value from
equation (IX.) in terms ofp^ , viz.,
which being substituted for /?, in the expression above, it becomes
and again, substituting for «^, x^, and 0, their several values above,
and reducing, we have
x(^ +^)( +/'.) hyper, log ^l^^^:^^!^^^
for the whole amount of the work done by the expansion of the
steam in the large cylinder, minus the resistance produced by the
reaction of the steam in the small cylinder ; whence, adding this to
the expression already obtained for the work done in the small
cylinder, and reducing, we obtain
. f hyper, log ^^f/^^J;^"^n)^a,/3,
which equals the actual work performed by Woolfe's engine, durin^^ "I
one stroke.
Then, in order to obtain the value of P, in this case the resistances^
against each square foot of the lar^r piston's surface, arising fron^^*
the load and friction of the engine, and from the imperfect vacuun^^^
in the condenser ; let w, /, and ^, represent the same quantities a ^^^
before, only taken per unit of the large piston's surface; thei— *^
we have
P = ^(l +^)+/+^,
which being multiplied by a, /,, equals the whole amount of rcr "
sistance developed during one stroke of the engine ; equal
Therefore, since where the engine has attained uniform motion the
resistance must equal the work performed, we have
Q
CHAP. VII.] THB nUU XNGIMB. 833
■h hyper, log ^^^^;^^\>^^^^n)%,/3
Then, if V equals the yelodty of the larger piston, and 9 the
V
nnmber of strokes which it makes per minute, we have » = j ,
*s
whence we obtain, for the value of j^^, from equation (XIV.)
* «iyV(/, +Xi) g'
Then, snbsdtuting this valae of />, in the above expression,
rednciug, and solving in respect of Q, V, and ; a, V, we have
«.Vf» + <rr> (! + »)+/+>]?
ll + hyper, log ^±i. + hyp«. log •«J''Jt*«>+"l^J
Q {^ + Hyper. U^ ;4il + hypT. U«^f±^4±fJ}
«.v £.
i + f
In order to put these formulsB under a more practical form, we
must snhstitute for the constants and coefficients their average values,
vis., for n, 7, and ar,as before, for condensing engines, /*» 125 lbs.,
9 as before, x, = 'OS/j, and x, = '05/,, then we obtfun
ie>«0Q{^24H hyper. log;^^hyper.U.^4,^^±S^}
^ " «, {2231S + •0029412 {)
(•,VS9D007SQ {•9624 + hyper, log ^±^'
334 THB STBAM RNOINB. [PART II.
The following table, giving the numerical value of the expression
hyper, log i±^+h.yper.logji^±^\±^) for jslues of
p varying from 77; to  , and for the three cases where — eqaals
4, 3 and 2, may sometimes be found useful : in this table the two
strokes are assumed to be of the same length, that is, /^ == Z^. .
(
Value of
Value of (hyper, log ^ + hyper. lo
«s((i + ^)+M,\
^«i(/, + ^.) + W
when a, « 4a).
when a, « 3a.
when a, = 2a,.
•10
316969
292674
257162
•15
2^88200
263906
2*28393
•20
265886
241591
206079
•25
247654
223359
187847
•30
232239
207944
1 72432
33
223901
199606
1 64094
35
218885
194591
159078
•40
207107
182813
147300
•45
196561
172277
136764
•50
187088
162793
127281
Sin^eacting Engine^ in which the Steam cuUa only upon the upper
surface of the Piston.
This class of steam engine is that usually known by the appella
tion of the Cornish pumping engine, for which purpose it is more
peculiarly adapted ; and from the surprising amount of work which
has been realized by means of it, with the consumption of a given
weight of fuel, has attracted a great deal of attention, and has become
almost universally adopted in mining districts.
As it is not our province here to enter into a detail of tbe
mechanical construction of these engines, we have only shown in
plate XII, figure 3, a section of the cylinder and valves ; it maj,
however, be remarked, that these engines exhibit the greatest me
chanical skill in their design and construction, and doubtless ooe
cause of the high duty which they perform results from the accuiacj
and precision with which the various parts are formed, and the
precautions adopted for preventing loss of heat by the steam io
passing through the engine.
In the plate, A is the cvlinder, which is surrounded by an outer
casing sufficiently large to leave a space of about an inch round the
whole of the exterior of the cylinder, which space (termed the steam
jacket) is always kept filled with steam of the same pressure as thtt
m the boiler, by means of the pipe B, which is in direct communici
CHAP. YII.] THB STEAM ENOINB. 335
tioD with the hoiler ; a similar space is also left helow the cylinder,
and kept constantly filled with steam. The ohject of thus sur
rounding the cylinder with hot steam, is to prevent its interior
surface from heing cooled during the upstroke; and for the same
purpose, the top of the cylinder, as also the upper part of the valve
box, and the entire steam pipe from the boiler to the cylinder, are
enveloped in an external case, the space between being filled with
sawdust or some other nonconducting substance.
The piston being at the top of the cylinder, the valve C is opened^
allowing the steam to pass from the boiler into the cylinder, where,
pressing on the upper surface of the piston, it causes its descent,
raising in so doing the pump rods and counterweight which are
attached to the other end of the engine beam. After the engine has
performed a portion of its stroke (varying from ^^ to ^ in
different engines), the valve C is closed, and the remainder of the
downward, or, as it is termed in Cornwall, the indoor stroke, is
performed by the expansion of the steam : the under side of the
piston is in communication with the condenser by means of the pipe
D, which is closed by the valve E towards the conclusion of the
downstroke. There being in the Cornish engine no crank, or other
means by which the precise length of the stroke is determined, it
becomes necessary to adopt some means of preventing the piston
descending too far, and by doing so injuring the bottom of the
cylinder ; the way in which this is effected, is by so regulating the
quantity of steam admitted through the valve C at the commencement
of the stroke, that it shall be just sufficient to carry the piston to the
proper distance, and no further; but since any variation in the
pressure of the steam in the boiler, or in ihe quantity of water
raised by the pump, would immediately destroy this adjustment, and
cause the piston either to fall short of or to exceed its proper stroke,
springs are placed so as to receive and stop the beam of the engine
when it descends too low, and at the same time a bell is rung, which
warns the enginedriver to lessen the quantity of steam admitted by
the valve C. The piston having been brought to a state of rest at
the bottom of the cylinder, the valve F is closed bv which the com
munication with the condenser is cut off, and the valve G, termed the
equUihrium vaive^ is opened, by which a free communication is effected
let ween the top and bottom of the cylinder, and an equilibrium esta
blished between the pressure on the upper and under side of the piston,
which, therefore, having now nothing to oppose its motion but the
friction of the engine, is drawn to the top of the cylinder by the
connt^^eight which it had raised during its downstroke, and which
is sufficiently heavy to raise the columns of water in the various
pmnp mains, and to overcome the friction of the engine. As, how
ever, the piston would continue its motion until it came into contact
with the cylinder cover, were it not checked, to do which the equi
librium valve G is closed before the piston has completed its up
stroke, after which the further motion of the piston, by compressing
the steam on its upper side and attenuating that below it, occasions a
386 THB ST£AM BKOINB. [PABT II.
Bofficient preponderance of pressure on its upper surface to bring it
gradually to a state of rest. The piston is now ready to resume its
downward stroke as soon as the valve C shall be opened, and the
steam from the boiler admitted to the top of the cylinder; this is
effected at the proper time by means of a contrivance termed the
cataract, by which the time which elapses between the up and down
strokes can be regulated at the pleasure of the person who baa charge
of the engine. The various contrivances by which the valves are
opened are not shown in the plate, as it would require too long a
description to render their use clearly intelligible ; it is sufficient to
remark, that the valve C can be closed at any portion of the stroke
which is desired, by means of a simple adjustment ; that the exhaus
tion valve F is so arranged as to close somewhat before the equi
librium valve G is opened ; that the injection cock of the condenser
is only open during the same time that the exhaustion valve F is
open ; and that the valve F is opened somewhat before the valve C,
in order that the steam beneath the piston may be perfectly con
densed as soon as the down stroke of the engine commences.
We will now proceed to deduce the formula for calculating the
effects of these engines, but since the nature of the work performed
by the Cornish engine during its indoor and outdoor strokes are
so essentially different, it will be necessary to investigate each
separately.
First, then, during the indoor or down stroke, we have the pres
sure of the steam direct from the boiler upon the upper surface of
the piston during the first portion of its stroke, which pressure will
be so little less than that in the boiler itself, that without any appreci
able error it may be assumed to be the same, for the piston, engine
beam, pump rods, and counterweight, comprising a joint mass weigh
ing several tons, require considerable force to put them in motion,
and only acquire that motion very gradually, so that the steam has
time to attam in the cylinder the same density and pressure as it
had in the boiler; and during the after part of the stroke, we have
the force due to the expansion of the steam above the piston. Now,
if we put a for the area of the piston, x for the length of the cylmder
equivalent to the clearance ana steam passages, p. for the pressure
in the boiler, l^ for the portion of the stroke pertormed before ex
pansion, and l^ for the whole length of the stroke from its com
mencement, we have from formula (XL*)
for the whole amount of work performed during the indoor stroke
of the engine; or, indeed, during both the indoor and outdoor
strokes, for the latter is, as already explained, performed by the
descent of a counterweight which was raised during the indoor
stroke ; and since the work expended in raising it is precisely equiva
lent to that which it performs during the outdoor stroke, the
amount which represents it would have to i^pear first on one ode of
CHAP. VII.] THE 8T£AM BNOINB. 387
the equation as a resistance, and afterwards upon the other as work
performed, we shall therefore simplify the calculation hy omitting it
altogether.
Now, the actual resistances to which the Cornish engine is ex
posed are as follows; viz., during the indoor stroke* the pressure on
the under side of the piston arising from the imperfect vacuum,
which we will as hefore designate hy v; the friction proper to the
motion of the engine when unloaded, equal /*; and the additional
friction caused hy the load of the engine, represented by ^ ^j during
the down stroke, and ^^^ during the up stroke, ^ having a different
value in consequence of the different load upon the engine, and ^
being the effective load measured per square foot of the piston's
surface: there is also a slight resistance occasioned by the steam
pressing with rather more force upon the upper surface of the piston
than the lower, in consequence of its having to pass through the passage
and equilibrium valve, but this difference of pressure is too small to re
quire being taken notice of; the compression of the steam, however,
in the upper part of the cylinder at the conclusion of the outdoor
stroke, and after the equilibrium valve has been closed, is of more
consequence, and requires to be considered. If, then, we put p^
for the pressure of the steam on each side of the piston at the
moment that the equilibrium valve is closed, p^ for the pressure of
the steam above the piston at the conclusion of the outdoor stroke,
and /, the length of the stroke performed after the valve is closed^
we have from equation (IX.),
or, sabstitnting for />,, in the last equation, its value from the first.
{,^(^)}7
Then, if / represent the length of the piston's outdoor stroke
remaining to be performed at any moment, and p the pressure of
the steam above the piston at the same moment, we have from
equation (IX.),
tnd for the work performed in the elementary space A/, we have
apAly
in which, substituting the above value of /?, and taking the integral
between the limits of / ss 0> and / =: Z,, we have
z
338 THE 8TBA1I BNOINE. [PABT II.
p^ dl in \ n /•»
= ax(^ +>'0 ^^^'' ^^^^1( 7*''
which hccomee, hy Bubstitating for p^ its value above in terms of/?,,
which expression equals the work required to be done by the piston
in compressing the steam in the upper part of the cylinder, after the
equilibrium valve is closed. But a portion of this work is perfonned
by the expansion of the steam on the under side of the piston, and
therefore we must deduct this from the above. Now, the pressure
of the steam on the under side of the piston at any moment being
represented by />, and the length of the piston's stroke from its
commencement at the same moment b^ng represented by I, we have
and for the work performed in the elementary portion of the stroke
represented by A /, we have ap A /, in which, substituting the abov&
value of py and integrating between the limits of I z= l^^ and^
/ = /j "— ^3> we obtain
r substituting for p^ its value above,
a(/.^,HX),fJ^,(; 4;>.)hyper. log ^±^±+1.!^
equal the work which the steam below the piston performs by ^l_ ts
expansion after the equilibrium valve is closed; and therefore t^^fc®
resistance actually to be overcome by the engine equals
i
CHJU>. VII.] THB 8TXAM BNOINB. 889
Therefore, the total resistance during both strokes of the piston,
equals
2Pa/, = a4 {e(l + (p, 4. (p,) + T + 2/)
 (/,  /« + A) hyper, log ^i^i^ J ;
and therefore, from the necessary equality between the work and
the renstance, we have
= a/,{e(l +*, + «),) + »+«/}
(/,/3 + X) hyper. Iog^A+i_J.
Now, since the Cornish engine only performs work in pumping
during its indoor stroke, it is usual to take its effective Telocity
as y/g = V, v being the number of double strokes which the engine
mi^es per minute. Then, the quantity of steam actually consumed
by the engine each double stroke, is only that which remains below
the piston when the equilibrium valve is closed, and which equals
a (/, — /, + X) ; and further, since the quantity of steam generated
eqoak that consumed, we have
1 V
m wbich v, = , and y =  ; making which substitutions
and solving in respect ofp^^ we obtain
wUeh latter expression is the value of j^,, alreadv given at page 887.
Then reducing, and solving in respect otp^^ we have
_ /,Q /, i2X n
^»a^V(/,/3 + X) /, + x 7
z 2
340
THB 8TBAM BNOINB.
[PABT II.
to
a
'3
a
'^
2
I
a
►
'So
I
08 O
If
•I?
•SIS
p
CO
'■s
2
•a
8l«
+
+
I:
+
+
>
^
+
II
Of
?
I: I:
+
I I
+
^
/<
/<
04
04
+
+
+
04
•^
*J^
^*
Is
+
9
/<
/<
X
+
/<
+
/<
•f
1
1
1
**•
t
t
^«
o
2
o
^
p
i
,£
3
j:
3
II
g^
+
I
s
04
+
Is
>
I
+
or
H
>
CRAP. VII.] THE 8TBAM BNOINE. 341
Singleacting Engine^ in which the steam presses only upon the
lower surface of the piston.
The engines belonging to this class are better known under the
title of atmospheric engines, from the down stroke being performed
entirely by the pressure of the atmosphere upon the upper surface
of the piston.
Plate XII, figure 4, is a section of the working cylinder of an
atmospheric engine ; the steam at a low pressure enters the bottom
of the cylinder A, by the valve B, and assists the counterweight at
the other end of the beam in raising the piston. When a certain
portion of the stroke has been completed, the valve B is closed, and
the counterweight being insufficient of itself to overcome all the
resistances of the engine, its velocity becomes gradually diminished,
and the piston at last comes to rest at the top of the cylinder. The
valve C is then opened, by which a communication is effected with
the condenser; and a vacuum being formed under the piston, the
pressure of the atmosphere upon its upper surface causes it to
descend. In order to prevent the piston making too long a down
stroke and coming in contact with the bottom of the cylinder, the
valve C is closed some time before the conclusion of the stroke, and
the oncondensed steam then remaining in the cylinder being com
pressed below the piston, gradually brings it to a state of rest.
The work performed by the steam during the up stroke will be
represented by formula (XL*) and if we put n for the pressure of
the atmosphere, that performed during the down stroke will be
lla/g, therefore we have
« (^ + Pi) {^1 + (^1 + X) hyper, log ^±^^  %/, + na/^
for the whole work performed during both strokes of the engine.
Then we have for the resistance resulting from the friction of the
engine when moving without any load, 2/2/9 and for the additional
friction caused by the load, together with the resistance of the load
itself, /^eO +?i + ^s)' ^^^9 finally, for the resistance caused by
the imperfect vacuum, if we put l^ for the length traversed by the
]nston after the valve C is closed, we have for the resistance before
the closing of the valve, a n (/j — l^) ; then if we put p for the
pressure of the uncondensed steam, at any moment when the piston
18 at a distance equal to / from the end of its stroke, we have from
equation (IX.),
l\\\q ' "J q
whenee the work to be performed will be represented by
^_ /3f X /n \ n
•<'.<fTr(i)"l/^'
= a(/,+X)(^ + ir) byper. log ^±^_2a/,.
« (^3 + ^) f  + » j hyper, log
342 THB BTBAM ENGINE. [p
Therefore, collecting these several resistances, we have
T w I uypcr. lUK —
la/, +a^,{f(l+^, +f,)
+ a* (/,/,) = a ( ^ + jP. ) /, + (/, +'X) hyper, log ^
Now, in order to find the quantity of steam consnmed
engine, we most deduct the quantity left below the piston w!
yalVe C is closed, from the quantity admitted through the '
during the early portion of the stroke; the latter equals a{
with Uie pressure p^, and the former a (/^  X) with the prea
then, in order to find the space (equal S) which it would oc
tha pressure />, we have from equation (IX.)
and solTing in respect of 8^ we have
B = a(/, + X)!L±i^.
And therefore the quantity consumed per minute equals
,a{(/.+X)(/,.X)l±i^} = Q,.;
then, substituting for v and v^ their yalues as before, redact
solving in respect of />j, we have
'^'~«yv(/. + x)^/j.x\,, ^ '; ?'
Then, substituting this value of p^ in the preceding ei
reducing, and solving in respect of V, Q, and f a V, we obtain
GHAP. Vll.] THE STEAM ENGINE. 343
V
<<.v
•"*'.+«{''¥(=+')[,Ti.+''— mi!
' Tbe remark made above with regard to the Cornish engine, of the
want of experiments from which to deduce the values of the several
resistances, will equally apply to tbe atmospheric engine, and we are
therefore obliged to leave the above formula in their general form.
PRACTICAL APPLICATION OF THE FOREOOINO FORMULA.
In the preceding investigation we have deduced for each engine
three forms of expression, giving the velocity of the piston in feet
per minnte or V, the quantity of water in cubic feet evaporated by
the boiler per minute or Q, and the effective useful work performed
by the engine per minute expressed in pounds raised through a
height of one foot or ^aV ; it may, however, be desirable, before
concluding the subject, to deduce ^om these, expressions for the
horse power, and t¥e quantity of coals consumed.
In estimating the horse power of a steam engine, it is usual to
assume the power of a horse to be equivalent to raising 33,000 lbs.
through a height of one foot in. a minute; and, therefore, to derive
the horse power of an engine whose effective useful work per minute
equals ^aV, we have
^gy _^ J the number of horses' power
33000 ""I of the engine.
Then, if we put F for the weight in pounds of coal, which is
required to evaporate a cubic foot of water, we have
^ r the number of pounds of coal consumed
^ I by the engine per minute,
QF J the number of pounds of coal consumed
and 33000 —  =  p^^ horse power.
The following table contains the value of F for four different
varieties of coal, compared together by Mr, Wicksleed.
344
THB STEAM BKOINB.
[part II
Description of Coal.
Na of Ibf . of Coal
required to evaporate
one cubic foot of
water «» F.
Welsh coal, used in Corawall . .
Best ditto, used at Old Ford . . .
Anthracite, ditto . . .
Best Newcastle small, ditto . , .
Derbysliire coals, ditto . . .
67414
65838
69337
73322
92292
TABLE I.— Quantity of Coals equivalent to the horse power or
33,000 lbs. raised one foot per minute in high pressure steam
engines, when the greatest possible effect is obtained *.
2S4 5
251
275
21*28
307 7
3202
343*1
Is
45
60
m
180
210
^11
II
74
U'B
297
445
142
104
Quant itr *nf maJ equfrAksi ta
ooe bone potter*
Foun<ls rai&fil otic fopi hi^h «^i^
r Aleut to th^ Immediate pover of
When working
■I full pivHiire,.
480
1*>3
08
82
74
70
6S
143
77
&0
51
48
414
mt fulL pi^MUFe.
llB.
2,70a,ooo
8,200,000
13,700,000
ie;,@oo,ow
18,000,000
10,200,000
20,BOO,OQO
Wbeti «oTlU.fi
9,30a,000
17,700,000
33,7041,000
26,2i>O,OO0
2g,704i,a(ia
32,200,000
k
TABLE IL — Quantity of Coals equivalent to the horse power om
33,000 lbs, raised one foot per minute in condensing steam engines^
when the greatest possible effect is obtained.
t
c^
ll^
PcKiQdf raiwd one fofil hift* egml
^
s. g
^M s Qtiant ity of €Oft] eaaivaJcnt to
¥sk9tt to thfr tUttwft'Btr paver (x^
^mcr^
fl
ll
£ *B. <iOm hone
power.
thw cteim produ
Efil
»ct by 8« lli^ OM
^ Mif
tL
Wh*n acbog
1 S worn warning
^ §'3 9.% full fireMUre*
WhffD iForiLlnia
H
Eipuultely^.
Bt Ml proiun.
npuulrtJjt
i
II1C}1I?I.
ItM.
Itii.
Ibt.
Ibi.
Ibi.
2^0
35
25
63
40i
2l,<KM3,0OO
33,100^000
"* J
2^45
45
74
63
38*
21,400,000
35,200,000
2fil'0
00 1
148
60
35!
22,400,000
37.500,000
^ /
275i)
»0 1
207
331
40.000,000
/
292 8
120
445
32i
41,000,000
/
307 7
150^ 1
593
32
42,400,000
/
a^2
ISO
74 2
314
42,700,000
/
3436
240
104
at
43,500,000
J
* The tables here given, marked I. II. III., were extracted, with the av
thor*! peroiisrion, ^m Mr. Tredgold*8 work on Railroads.
CHAP. VllI
THE 8TB1M ENOINB.
345
Remarhon Tobies I. and II. — The colamns showing the pounds
in engme oug^t to raise one foot high, by the heat of one bnshel of
coals, are added chiefly for the pnrpose of comparison with actual
practice. Now, it is stated, that after the most impartial examina
tion for seTeral years in succession, it is found that Woolf's engine,
at Wheal Abraham Mine, raised 44,000,000 pounds of water, one
foot high, witb a bushel of coals. And, ^^ the burning of one bushel
of good Newcastle or Swansea coals in Mr. Watt's reciprocating
engines, working more or less expansively, was found, by the
aeconnts kept at tbe Cornish mines, to raise from 24,000,000 to
32,000,000 pounds of water one foot high; the greater or less
effect depending upon the state of the engine, its size, and rate of
working, and tbe quality of the coal."
We shall further add the results of half a year's reports taken,
without selection, from Lean's Monthly Reports on the work per
fonned by the steam engines in Cornwall, with each bushel of coals.
The numbers show the pounds of water raised one foot high with
etch bushel, from January to June, 1818.
Name of
Pounds raised one foot by a bushel of coals.
January.
February.
March.
April.
May.
June.
Stotf ComnMo Bb
WW VOT <Woolf%
^WtedAbnlMm (ditto)
.Ditto (ditto)
■Whad Unity (ditto)
'Daloomh Cneine
WlMl Abraham Engine
iUiilsd Mines Cngj
iTicskiiby Encine
[WhsalCfaScgEn
Engine
>J8B,000
ao.834,000
41JM7.000
314X10.000
42^89,000
aa,sw,ooo
38.306,000
38,733.000
98,406,000
SS.4S4,000
26,108.000
3S.364,000
98,000.000
393)6.000
41.364.000
36,180,000
31.830,000
30,375.000
38,319.000
91,808,000 29,969.000
99,611,000 26,064.000
23,606.000
23,836.000
30.445,000
96.978,000
40.490.000
35,715.000
31.427.000
41.867.000
33.594.000
39.723.000
23.e86.r"
,000 29,
29,032.000 30.336.000
31 .520,000 '34.352,000
1,702.000 '34.846.000
41.888.000
33,934,(NN)
33.564,000
41.823.000
33,932,000
38.233.000 '38.143.000
33.714.000 ,34.291 .0(N»
33.967.000 '30.105.000
40.615.000 '42,098.<NIO
— 35,797.000
Tliese numbers are less than the immediate power of the engines,
bj the friction and loss of effect in working the pumps ; hence, in
comparinff them with Mr. Tredgold's table, it may be inferred that
be msde bis calculations from such data as can be realised in practice.
It is known from experience, that a cubic foot of water can be con
verted into steam equal in force to the atmosphere, with 7 lbs. of
Newcartle coals; but we also know the attention necessary to pro
dace that effect, and therefore have assumed that %f^ lbs. will be
wjoired for that purpose.
Aeeording to Mr. Leans Monthly Report, for January, 1 833, the
foDowing engines raised more than 50,000,000 pounds, one foot
* r consuming % bushel of coals : —
846
THE 8TBAM SNOINB.
[PABT II.
Name of Mine.
Diameter of
cylinder.
Pounds raised
bya
bushdofoMls.
Strokes
Cardrew Downs . . .
Binner Downs . . . .
Ditto
Consolidated Mines . .
Ditto
Ditto
Ditto
Ditto
Ditto
Polgooth
Pembroke
East Crinnis
Wheal Leisure . . . .
Wheal Vor
Ditto
Ditto
Poladras Downs . . .
Great Work
Wheal Towan . . . .
Ditto
Wheal Falmouth Consols
Wheal Darlington . . .
66 inches
70
65
90
70
65
90
90
65
66
40
76
66
80
80
53
70
60
80
80
70
80
single
51,831,751
57,942,435
55,931,852
51,713,913
61,846,133
54,726,957
59,978,983
52,040,672
65,617,011
70,240,452
61,170,237
62,097,533
53,506,372
88,504,900
65,471,147
53,938,177
56,766,668
65,460,248
73,159,628
68,782,390
54,334,137
66,058,518
734
1013
850
486
593
8*60
4*69
650
528
800
667
704
358
622
607
637
878
608
490
698
622
847
Of the above, the engine of greatest operation, the first at Whesl
Vor, raises the water 190 fathoms, ^i seven lifts, drawing perpen
dicularly 160 fathoms, and the remainder diagonally. Main beam
over the cylinder ; stroke in the cylinder ten feet ; one balancebob
at the surface, and three under ground.
The following extract from Leans Monthly Report, for June 1841,
will show the advance which had been then made.
Name of Mine and Engine.
Wheal Darlington, Eastern . .
Great Wheal Fortune, Wheal 1
Prosper ........ J
Duffield
WhealJulia. . . .
Godolphin, Sims* . .
Great Work, Leeds* .
Wheal Vor, Borlase's .
Ditto, Trelawny*s
North Roskoar, New .
Wheal Unity Wood, WUliams's
HallenBeagle
Diameter of
cylinder,
in indies.
80, single
80
80
80
60
80
80
70
80
70
Poundi lifted
one foot bra
bushel of ooala.
No.ofsti6kci
perminut&
81,681,776
976
64,169,466
646
64,438,341
668
60,966,983
677
60,166,186
630
66,776,208
670
74,927,176
690
60,634,127
6KK>
67,364,238
476
61,168,649
664
62,314,765
660
CHAP. VII.]
THB STBAJi BNOINB.
347
Nun* oi Mine «iid Ea^am.
D1uD«leror
Pftundi lifted
Ddv foot by »
buBbti<frcc*Ift.
^fo.flflt^a1tM
pcrmLtiiitew
Wheal UnitT
70, tingle
7a M
50, combined
85, kitig]«
m> „
30 „
B5 „
85 „
50 ..
ao „
80 „
66 „
08,462,34 fi
70,(J35,787
88,096,178
9«,659,570
55,219,358
61,652,653
€7.044, 127
a,i3i,7ei
54,477,451
59,2tf7,244
77,8M,927
71,144,002
670
360
430
400
B40
1030
7i)7
660
365
436
530
5W
Catq Br«ft . , , . .
INtto, Siou' . * *
United Miiiei, Tajlori' *
DitUi, OurdoEo'i .
Ditto, Elfbn's ,
Ditto, Loam'i . ,
Ditto, Hocking'i
United HilJif WiUiami^f .
Fowef CcHiaolfj Aa»tin*f .
PfllirMitll
TABLE in.— /S^oiriny r^ gfecto of a force of traction of 100 Ibt.
at different velocities^ on Canals^ Railroads^ and Turnpikeroads *.
{From Tredgold,)
TdodlTOflfcieloA.
Lo«d nwjvtd by « power of 100 llw
•HBIlll.
On » C»a»l
On » l»Td lUiliny,
Onftln«l
TuTDpDw Roadi
mus
DTMved,
Total
ErH>vddt
tllfert.
Tot*I
IDAtl
r
5
6
1
8
10
135
366
440
513
1 686
733
8^80 '
1026
11 73
1320
1466
100
Ibi.
55,500
38,542
Si8,310
21,060
,13,875
9,635
7,080
5,420
3,468
1,900
Ibt.
39,400
27,361
20,100
15,890
9,850
6,840
5,026
3,848
3,040
2,462
1,350
Itat
li,400
14,400
14,400
14,400
14,400
14,400
14,400
14,400
14,400
14,400
14,400
lb*.
10,800
10,800
10,800
10,800
10,800
10,800
10,800
10,800
10,800
10,800
10,800
1,800
1,800
1,600
1,800
1,800
1,800
1,800
1,800
1,800
1,800
U800
1,350
1,350
1,350
1,350
1,350
1,3150
1,360
1,350
1,360
1,350
1,350
This table is intended to exhibit the work that may be performed
by the same mechanical power, at different velocities, on canals, rail
roads,' and tampikeroads. Ascending and descending bj locks or
canals^ may be considered equivalent to the ascent and descent of in
clinations on railroads and tampikeroads. The load carried, added to
the weight of the vessel or carriage which contains it, forms the total
• Though the force of traction on a canal varies as the square of the velocity,
the me c h anical power necessary to move the boat is usually reckoned to increase
as the cube of the velocity. On a railroad, or turnpike, the force of traction
is constant; but the mechanical power necessary to move the carriage inereases
as the vdodty.
348 8TRBN0TH OF ANIMALS. [PART II.
mass moved ; and the useful effect is the load. To find the effect on
•canals at different velocities, the effect of the given power at one
velocity being known, it will be as 3' : 2*5^ : : 55,500 : 38,542.
The mass moved being very nearly inversely as the square of the
velocity ; at least, within certain limits.
This table shows, that when the velocity is five miles per hour, it
requires less power to obtain the same eflfect on a railway than on a
eanal ; and the lower range of figures is added to show the velocity
at which the effect on a canal is only equal to that on a turnpike
road. By comparing the power and tonnage of steam vessels, it will
be found that the rate of decrease of power by increase of velocity,
is not very distant from the truth ; but we know that in a narrow
canal the resistance increases in a more rapid ratio than as the square
of the velocity *, that is, within certain limits ; beyond them, there is
a remarkable change in the circumstances of resistance.
Sect. IV. Animal Strength as a Mechanical Agent.
1. The force obtained through the medium of animal agency, evi
dently varies, not only in different species of animals, but also in dif
ferent individuals. And this variation depends, first, on the particular
constitution of the individual, and upon the complication of causes
which may influence it ; secondly, upon the particular dexterity ac
quired by habit. It is plain, that such a variation cannot be sub
jected to any law, and that there is no expedient to which we can
have recourse but that of seeking mean results.
Secondly, the force varies according to the nature of the labour.
Different muscles are brought into action in different gestures and
positions of an animal which labours ; the weight itself of the ani
mal machine is an aid in some kinds of labour, and a disadvantage in
others ; whence it is not surprising that the force exerted is different,
in different kinds of work. Thus the force exerted by a man is
* Aooording to the interestinff researches of Du Baat, the retistaDoe to the
motion of boats, even in canals, may be regarded as proportional to the
square of the velocity, or R as V* neariy, provided R be made to depend upon
the transverse sections of the vessel and the canal in which it movea. If c be
the vertical section of the canal, and b the vertical section of the immersed por.
tion of the boat, or barge ; then
The mean of Dn Buat'i ezperimmiu givM, K = 8'46, or
^  846 '
but these experiments were not so numerous and varied as might be wished.
See PrincipeM d* HpdrauHque^ tom ii. pp. 340, 342, &c.
CHAP. VII.] 8TRBNGTH OF ANIMALS. 349
different, in carrying a weight, in drawing or pushing it horizontally,
and in drawing or pushing it vertically.
Thirdly, the force varies according to the duration of the lahour.
The force, for example, which a man can exert in an effort of a few
instants, is different from that which he can maintain equahly in a
course of action continued, or interrupted only hy short intervals, for
a whole day of lahour, without inducing excessive fatigue. The
former of these may he called Absolute Force j the latter Permanent
Force, It is of use to hecome acquainted with them hoth, as it is
often advantageous to avail ourselves sometimes of the one, some
times of the other.
Lastly, the force varies according to the different degrees of velo
city widi which the animal, in the act of labouring, moves either its
whole body, or that part of it which operates. The force of the
animal is the greatest when it stands still ; and becomes weaker as
it moves forward, in proportion to its speed ; the animal acquiring,
at last, such a degree of velocity as renders it incapable of exerting
any force.
2. Let ^ be a weight equivalent to the force which a man can
exert, standmg still : and let V be the velocity with which, if he pro
ceeds, he is no longer capable of exerting any force : also, let F be a
weight equivalent to the force which he exerts, when he proceeds,
equably, with a velocity v.
Then F will be a functiou of v, such that, 1st, it decreases whilst
V increases ; 2nd]y, when » = 0, then F = ^ ; 3rdly, when v = V,
F = 0.
3. Upon the nature of this function, we have the three following
suppositions.
1. Fs^fl— V (Bouguer, Man. dee Vais.)
2. Feaf^i—ljY (Euler, Nov. Comm. Pet. tom. IIL)
3. Frs^Tl— M. (lb. tom. VIII ; and Act of Bowers.)
4. CoroH. 1 . The effect of the permanent force being measured
by the product F v, the expression for the effect will be one of the
three following, accordingly as one or other of the suppositions is
adopted.
1. Fv(l),or^«(l.).
^ ^^ a/(' !)'"*"(' S)
3. Pv(l.^!),or^. (l^y.
350
ANIMAL 8TRBNQTH : SCHULZB's BZPBRIMBNT8. [PART II.
5. Coroil. 2. To know the weigbt with which a man shoold be
loaded, or the velocity with which he onght to moTe, in order to pro
dace the greatest effect, we most make ^ . Fo e=s 0.
Whence we shall haye
1. F =  ^; and v =
2
2. F =  ^ ; and v =
3.
4
>/3
and I? =  V.
3
V = 05773 V.
6. CoroB, 3. And the yalae of the greatest effect, according to the
seyeral hypotheses, will be.
1. \^y:
2.
3 >/3
f V s 03836 f V :
3. 27*^
Bat which of the three suppositions ought we to prefer ? And sre
we certain that any of them approximates to the true law of nature?
Mr. Schuize made a series of experiments with a view to the de
termination of this point*, and with regard to men decided in favour
of the last of Euler s formulae : viz.
,=,(. 1).
As the experiments of this philosopher are very little known in
England, I shall here present hss brief account of them.
7. To make the experiments on human strength, he took at rao*
dom 20 men of different sizes and constitutions, whom he measured
and weighed. The resnlt is exhibited in the following table.
Order.
Height.
Weight.
Order.
Height
Weight
1
5/ 3// 4W
122
11
5/ 9// 7///
132
2
6 2 3
134
12
6 1 4
157
8
6 7 2
165
13
6 8 2
175
4
6 6
181
14
5 4 1
117
5
6 11 2
177
16
5 10 8
192
6
6 4
158
16
6 8
133
7
6 8 3
180
17
4 11 2
147
8
6 2 1
117
18
6 8 9
124
9
6 4 8
140
19
5 6
163
10
6 4
126
20
6 10 1
181
* Mem. Acad. Sdcnc. Bsrlin, for 1783.
V^ll.] ANIMAL araKNOTH: SCHULZB's BXPBRIMENT8.
851
e the heights are expressed in feet (marked ^), inches C')y
m ("')y the feet being those of Rhinland, each 1235 English
The weights are in pounds, which are to our ayoirdupois
30 to 29.
ind the strength that each of these men might exert to raise a
yerticaily, Mr. Schnlze made the following experiments : —
took Tarious weights increasing bj 10 lbs. from 150 lbs. up to
B. ; all these weights were of lead, having circular and equal
To use them with suodess in the proposed experiments, he
the same time a kind of bench made, in the middle of which
hole of the same size as the base of the weights : this hole was
y a circular cover when pressed against the bench ; at other
t was kept at about the distance of a foot and a half above the
bj means of a spring and some iron bars. To prevent the
. with which this cover was loaded during the experiment from
; down the cover, lower than the level of the surface of the
he had several grooves made in the four iron bars, which sus
the cover, and which at the same time served to hold up the
at any height where it might arrive by the pressure of the
I as soon as the pressure of the weight ceased.
Mr having laid the 150 lbs. weight on the cover, and the other
s in succession increasing by 10 lbs. up to 250 lbs., he made
[lowing experiments with the men whose size and weight are
ibove, by making them lift up the wights as vertically as pos
II at once, and by observing the height to which they were
i lift them. The annexed table gives the heights observed for
Fereat weights mariced at its head.
BO
IGO
170
180
194
200
iio
223U 230
240
MO
m
*f m
it m
ri m
jtf m
U Hi
it tit
** H* if aif'i
ff Iff
B 4
4 11
4 4
a a
2 B
1 1
I
10
6 6
5 7
4 7
3 u
2 5
6
17 3
7 3
6 5
a 9
4 11
4
3 ,
3 8 3 1
1 4
7 6'
7 2
d 10
5 3
4 7
4
4 7 3 2
1 3
M 1
9 7
8 5
7 10
7 1
5 10
6 6 4 1
1
U
13 5
12 8
n
10 I
8 a
3 8 I 11
2
11
11 3
10 5
9 3
8 1
6 9
& 3
6 J 3 2
1
10 s
4
8 11
a 1
« 11
h SO
8 3
7 1
5 fl
4 I
3
1 3
« b
4 7
3 g
2 5
1 7
4
B table proves that the size of the men employed to raise the
ts Terdcally has considerable influence on the height to which
ronght the same weight. We find also that the height dimi
in a much more considerable ratio than the weight increases ;
e may therefore conclude, that it is advantageous to employ
nen when it becomes necessary to draw vertically from below
ds : and on the contrary, it is more advantageous to employ
>f a considerable weight, when it is required to lift up loads by
352
ANIMAL STBBNOTH: SCHULZE's EXPERIMENTS. [PABT II.
means of a pulley about which a cord passes, that the workmen may
draw in a vertical direction, from above downwards. To find the
absolute strength of these men in a horizontal direction, Mr. Schulze
proceeded thus : —
Having fixed over an open pit a brass pulley, extremely well made,
of 15 inches diameter, whose axis, made of well polished steel to
diminish the friction, was ^ of an inch in diameter, he passed over
this pulley a silk cord, worked with care, to give it both the neces
sary strength and flexibility. One of the ends of this cord carried a
hook to hang a weight to it which hung vertically in the pit, whilst
the other end was held by one of the 20 men, who in the first order
of the following experiments made it pass above his shoulders;
instead of which, in the second, he simply held it by his hands.
Mr. Schulze had taken the precaution to construct this in such a
manner that the pulley might be raised or lowered at pleasure, in
order to keep the end of the cord held by the man always in a hori
zontal direction, according as the man wss tall or short, and exerted
his strength in any given direction.
He had made the necessary arrangements so as to be able to load
successively the basin of a balance which was attached to the hook
at the end of the cord which descended into the pit, whilst the man
who held the other end of this cord employed all his strength with
out advancing or receding a single inch.
The following tables give the weights placed in the basin when
the workmen were obliged to give up, having no longer sufficient
strength to sustain the pressure occasioned by the weight. To pro
ceed with certainty, Mr. Schulze increased the weight each time by
five pounds, beginning from 60, and took the precaution to make
this augmentation in equal intervals of time ; having always precisely
a space of 10 seconds between them. The result of these observa
tions repeated several days in succession, is contained in the follow
ing tables.
I. When the cord passed over the shoulders of the workmen :
Order.
lbs.
Order.
lbs.
Order.
Ibe.
Order.
lU.
1
95
6
100
11
95
16
95
2
105
7
115
12
100
17
100
3
110
8
105
13
110
18
90
4
100
9
95
U
90
19
100
5
105
10
90
15
110
20
100
CHAP. VII.] ANIMAL STRENGTH: MEN.
II. When the cord was simply held before the man :
353
Order.
Ibt.
Order.
lbs.
Order.
lbs.
Order.
lbs.
1
90
6
100
11
90
16
90
2
105
7
no
12
90
17
90
3
105
8
100
13
100
18
85
4
90
9
90
14
85
19
100
5
95
10
85
15
105
20
100
These two tables show that men have less power in drawing a
cord before them than when they make it pass over their shoulders ;
thej show, also, that the largest men have not always the greatest
strength to hold, or to draw in a horizontal direction, by means of a
cord. To obtain the absolute velocity of these twenty men, Mr.
Schnlze proceeded as follows : —
Having measured very exactly a distance of 12,000 Rhinland feet,
in a plane nearly level, he caused these twenty men to march with a
fair pace, but without running, and so as to continue during the
period of four or five hours ; the following is the time employed in
describing this space, with the velocity resisting for each of them.
Order.
1
Time.
Velocj
Order.
Time.
Veloc
Order.
Time.
Veloc
4018
494 1
8
4009
499 !
15
3617
551
2
4112
485
9
4020
496 1
16
4128
482
3
3908
555 '
10
4051
490 '
17
4225
471
4
3940
504
11
3617
551
18
4019
498
5
3419
583
12
3811
524
19
3957
501
e
3511
568
13
3805
525
20
3751
529
7
3807
525
14
3701
540
It is necessary to mention, with regard to these experiments^ that
Mr. Schulze took care to place at certain distances persons in whom
he could place confidence, in order to observe whether these men
marched uniformly and sufiiciently quick without running.
Having thus obtained not only the absolute force, but the absolute
velocity also, of several men, he took the following method to deter
mine their relative force.
He made use of a machine composed of two large cylinders of
very hard marble, which turned round a vertical cylinder of wood,
and moved by a horse, which described in his march a circle of
10 Rhinlaod feet. This machine appeared the most proper to make
the subsequent experiments, which serve to determine the relative
strength that the men had employed to move this machine, and which
is used hereafter to determine which of Euler's two formulas ought
to be preferred.
To obtain this relative force, he took here the same pulley which
354 ANIMAL 8TRIN0TH : MBK. [PART II.
served in the preceding experiments, by applying a cord to the
vertical cylinder of wood, and attaching to the other end of this cord,
which entered into an open pit, a sufficient weight to give suceseivelv
to the machine different velocities.
Having applied in this manner a weight of 215 lbs., the machine
acquired a motion, which after being reduced to a uniform velocity,
taking into account the acceleration of the weight, of the friction,
and of the stiflfness of the cord, gave 2*41 feet velocity ; and having
applied in the same manner a weight of 220 lbs., the resulting
uniform motion gave a velocity of 2*47 feet. These two limits are
mentioned because they serve as a comjmrison with what imme
diately follows : Mr. Schulze began these experiments with a weight
of 100 lbs., and increased it by five every time from that number op
to 400 lbs.
He made this machine move by the first seven of his workmen,
placing them in such a way that their direction remained almost
always perpendicular to the arm on which was attached the cord
which passed over their shoulders in an almost horizontal direction.
Thus situated, they made 281 turns with this machine in two
hourH, which gave for their relative velocity p = 2*45 feet per second.
We have also the absolute force, or f, from these scTen men,
by the above table, ^ 730 lbs. : and their absolute velocity or
V = 530 feet.
Therefore, by substituting these values in the first formola, we
find the relative force F = 205 lbs., which agrees very well with
what we have just found above.
If instead of this first formula the second be taken, it gives
F =: 153 lbs., which is far too little.
By this it is evident, that the last of Ruler's two formulae is to be
preferred in all respects. Mr. Schulze made a great number of com
binations, and almost alu'ays found the same effect.
Dividing the 205 lbs. which we have just found by seven, the
number of workmen, we get 29 lbs. for the relative force with
2'45 feet relative velocity for each man, which is rather more than
the values commonly adopted in the computation of machinery. A
number of other observations on different machines have given the
same result; that is to say, we must value the mean human strength
at 29 or 30 lbs., with a velocity of 2feet per second.
To obtain the ratio of the strei^gth of a horse to that of a mm,
Mr. Schulze proceeded in a similar manner; but his results, in
reference to that inquiry, are neither so correct nor so interesting.
8. In the first volume of my Mechania^ I stated the ayerage force
of a man at rest to be 70 lbs., and his utmost walking velocity when
unloaded, to be about six feet per second; and thence inferred that a
man would produce the greatest momentum when drawing Sl^lbs*
along a horizontal plane with a velocity of two feet per second. But
that is not the most advantageous way of applying human strength.
9. Dr. Desagaliert asserts, that a man can raise of water or any other
OHAP. VII.] ANIMAL STRENGTH : MEN. 355
weight aboat 550 lbs., or one hogshead (weight of the vessel included),
ten feet high in a minute; this statement, though he says it will
bold good for six hours, appears from his own facts to be too high,
and 18 certainly such as could not be continued one day after another.
Mr. Smeaton considers this work as the effect of haste or distress ;
and reports, that six good English labourers will be required to raise
21141 cubic feet of sea water to the height of four feet in four hours:
in this case, the men will raise a very little more than six cubic feet
of freah water each to the height of 10 feet in a minute. Now, the
hogshead containing about 8^ cubic feet, Smeaton s allowance of work
proves less than that of Desaguliers in the ratio of 6 to 8^ or 3 to 4.
And as his good English labourers who can work at this rate are
ettiinated by him to be equal to a double set of common men picked
Qp at random, it seems proper to state that, with the probabilities of
voluntary interruption, and other incidents, a man's work for several
neoessive days ought not to be valued at more than half a hogshead
laised 10 feet high in a minute. Smeaton likewise states, that two
ordinary horses will do the work in three hours and twenty minutes,
which amounts to little more than two hogsheads and a half raised
10 feet high in a minute. So that, if these statements be accurate,
one horse will do the work of five men.
Mr. Emerson affirms, that a man of ordinary strength, turning a
roller by the handle, can act for a whole day against a resistance
equal to 30 IbSs weight ; and if he works 10 hours a day, he will
rsise a weight of 30 lbs. through 3^ feet in a second of time ; or, if
the weight be greater, he will raise it to a proportionally less height.
If two men work at a windlass, or roller, they can more easily draw
up 70 lbs. than one man can 30 lbs., provided the elbow of one of the
handles be at right angles to that of the other. Men used to bear
loads, such as porters, will carry from 150 lbs. to 200 or 250 lbs.,
scGording to their strength. A man cannot well draw more than
70 Iba. or 80 lbs. horieontally : and he cannot thrust with a greater
force acting horisontally at the height of his shoulders, than 27 or
80 lbs. But one of the most advantageous ways in which a man can
exert bis force is to sit and pull towards him nearly horizontally, as
in the action of rowing.
M. CSoulorob communicated to the French National Institute the
results of various experiments on the quantity of action which men
csn afford by their daily work, according to the different manners in
which they employ their strength. In the first place he examined
the quantity of action which men can produce when, during a day,
they mount a set of steps or stairs, either with or without a burden.
He found that the quantity of action of a man who mounts without a
burden, having only his own body to raise, is double that of a man
loaded with a weight of 68 kilogrammes, or 150 lbs. avoirdupois, both
eoDtinuing at work for a day. Hence it appears how much, with
equal fatigue and time, the total or absolute effort may obtain different
fdnes by varying the combinations of effort and velocity.
But ihe word effect here denotes the total quantity of labour
A A 2
356 ANIMAL STRENGTH: MBK. [PABT IT.
employed to raise, not only tbe burden, but the man himself; and,
as Coulomb observes, what is of the greatest ]mK)rtance to consider
is the useful effect,, that is to say, the total effect, deducting the value
which represents the transference of the weight of the man's body.
This total effect is the greatest possible when the man ascends without
a burden ; but the uteful effect is then nothing : it is also nothing
if the man be so much loaded as to be scarcely capable of movbg,
and consequently there exists between these two limits a value of tbe
load such that the useful effect is a maximum. M. Coulomb supposes
that the loss of quantity of action is proportional to tbe load (an
hypotliesis which experience confirms), whence he obtains an equa
tion which, treated according to the rules of maxima and minima,
gives 53 kilogrammes (117 lbs. avoirdupois) for the weight with
which the man ought to be loaded, in order to produce during one
day, by ascending stairs, the greatest useful effect : the quantity of
action which results from this determination has for its value 5(^ kilo
grammes ( 1 23 lbs. avoirdupois) raised through one kilometre, or nearly
1094 yards. But this method of working is attended with a loss of
threefourths of the total action of tlie man, and consequently costs four
times as much as work, in which, after having mounted a set of steps
without any burden, the man should suffer himself to fall by any means,
so as to raise a weight nearly equal to that of his own body.
From an examination of the work of men walking on a horizontal
path, with or \%nthout a load, M. Coulomb concludes that the greatest
quantity of action takes place when the men walk being loaded ; and
is to that of men walking under a load of 58 kilogrammes (128 lbs.
avoirdupois) nearly as 7 to 4. The weight which a man ought to
carry in order to produce the greatest useful effect,, namely, that effect
in which the quantity of action relative to the carrying his own
weight is deducted from the total effect, is 50*4 kilogrammes, or
111*18 lbs. avoirdupois
There is a particular case which always obtains with respect to
burdens carried in towns, viz., that in which the men, after having
carried their load, return unloaded for a new burden. The weight
they should carry in this case, to produce the greatest efiect, is
61*25 kilogrammes (135^ lbs. avoirdupois). The quantity of osefol
action in this case, compared with that of a man who walks freely to^
without a load, is nearly as 1 to 5, or, in other words, he employs to
pure loss ^ths of his power. By causing a man to mount a set of
steps freely and without burden, his quantity of action is at leait
double of what he affords in any other mediod of employing ^
strength.
When men labour in cultivating the ground, the whole qoantitj
afforded by one man during a day amounts to 100 kilogrammes ele
vated to 1 kilometre, that is, 220*6 lbs. raised 1094 yards. M.
Coulomb, comparing this work with that of men employed to csrr;
burdens up an ascent of steps, or at the pileengine, finds i loM
of about ^ th part only of the quantity of action, which may he
neglected m researches of this kind.
CHAP. VII.] ANIMAL STRBNGXH : IIBN. 357
In estimating mean results we should not determine from experi
ments of short duration, nor should we make any deductions from the
exertions of men of more than ordinary strength. The mean results
baye likewise a relation to climate. '' I have caused," says M. Cou
lomb, ^ extensive works to be executed by the troops at Martinico,
where the thermometer (of Reaumur) is seldom lower than 20^
or 77^ of Fahrenheit). I have executed works of the same kind by
\e troops in France, and I can affirm, that under the fourteenth
degree of latitude, where men are almost always covered with perspir
ation, they are not capable of performing half the work they could
perform in our climate."
10. Entirely according with these are the experiments of Regnier,
by means of a dynamometer, the results of which not only established
the superiority of civilized men over savages, but that of the English
man over the Frenchman. The following is reduced from one of
Regnicr's tables of mean results.
£
Strength.
Savages, of Van Dicmen's Land
New Holland . . .
Timor
Civilised men : French . . .
English . . .
With the
With the
hands.
reins.
Ibf. (m.
lbs. OS.
30 6
51 8
14 8
58 7
10 2
69 2
22 1
71 4
23 8
11. A porter in London is accustomed tu carry a burden of
200 Ibe. at the rate of three miles an hour; and a couple of chairmen
continue at the rate of four miles an hour, under a load of 300 lbs.
Yet these exertions, Professor Leslie remarks, are greatly inferior to
the subanltory labour performed by porters in Turkey, the Levant,
and generally on the shores of the Mediterranean. At Constan
tinople, an Albanian porter will carry 800 or 900 lbs. on his back,
stooping forward, and assisting his steps by a short staff. Such loads,
however, are carried for very short intervals. At Marseilles it is
affirmed that four porters carry the immense load of nearly two tons,
by means of soft hods passing over their hcads^ and resting on their
shoulders, with the ends of poles from which the goods are sus
pended.
12. With regard to the magnitude of the comparative efforts of
man in different employments, the late Mr. Robertson Buchanan
ascertained, that in working a pump, in turning a winch, in ringing
a bell, and rowing a boat, die dynamic results are as the numbers
100, 167, 227, and 248.
According to the interesting experiments described in M. Hachette's
Traiie des Mctckinesy the dynamic unity being Uie weight of a cubic
metre of water raisid to the height of one metre [that is, 2208 lbs.
avoirdupois, or 4 hogsheads raised to the height of 3*281 feet, or
356 ANIMAL STRSlfeTH: lfEK» [PABT
1*3124 hogsheads to the height of 10 feet], we have the folio?
measures, at a medium, of the daily actions of men.
Dyn. I
1. A man marching 7 hours on a slope of 7 degrees
with a load of from 15 to 18 lbs =
2. Marching in a mountainous country without load =
3. Carrier of wood up a ladder, his weight 123 Ibs^ his
load llTlbs =
4. Carrier of peat, up steps, his own weight compnced,
112 to =
5. Man working at the cord of a pulley to raise the ram \
of a pile engine: three examples )
6. A man drawing water from a well by means of a cord as
7. Man working at a capstan :=
8. Man working at a capstan to raise water, mean of 24
observations ! =
The unit of transport being the weight of a cubic metre of w
carried a metre (or 2208 lbs. carried 3281 feet) upon a horizo
road, we have for the daily action,
Dyn. T
1. A man travelling without load on a flat road, his weight
1 54 lbs. his journey 31 ^ miles ^ 3
2. A soldier, carrying from 44 to 55 lbs., travelling 12
miles, 1800 to = 1
3. Ditto, a forced march of 25 miles = 2
4. A French porter, weight of the man not included,
792 to =«
5. Porter with wheelbarrow, weight of the man not
included b: 1
6. Porters with a sledge ss
7. A man drawing a boat on a canal; 110310 lbs. coq
veyed 6 J miles * xr= 550
14. Mr. B. Bevan, an able engineer, has made experiments on
application of human energy to the use of augurs, gimlets, scr
drivers, &c. He has presented to the public the following list,
^)echnen ; premising that many ordinary operations are performe
a short space of time, and may therefore be done by greater exei
than if a longer time was necessary. Thus a person, for a a
time, is able to use a tool or instrument called
lbs.
A drawingknife, with a force of 100
An augur, with two hands 100
A screwdriver, one hand 84
A common bench vice handle 73
A chisel and awl, vertical pressure 72
A windlass, handle revolving 60
CHAP. VII.] ANIMAL STBBNOTH : MEN. 35^
lb«.
Pincers and pHers, compression 60
A handplane, horizontallj 50
A hand or thambvice 45
A handsaw 36
A stockbit, revolving 16
Small screwdrivers, or twisting by the
thumb and forefinger only 14
15. M. Morisot informs us that the time employed by a French
stonemason's sawyer, to make a section of a square toise (40*89
square feet English) in different stones, is as below : viz.
hours.
Calcareous stone, equal grain, spec. grav. 2000 45
hard, spec. grav. 2300 62
Liais, ditto hard, fine grain, spec. grav. 2400 67
Pyrenean alabaster^ the softest of the marbles 56
Normandy granite 504
Granite from Vosges 700
Red and green porphyry 1177
The workmen ordinarily made 50 oscillations in a minute ; each
stroke about 15 inches.
16. Hassenfratz assigns 13 kilogrammes as the mean effort of such a
man ; but M. Navier, in his new edition of BdidoTy Architecture Hydrau
Uque^ regards this estimate as too high. If Hassenfratz were correct,
the daily quantity of action of the sawyer would be equivalent to 376
kilogrammes elevated to a kilometre (or 818 lbs. raised g of a mile),
a quail titv more than triple that of a man working at a winch. M.
Navier gives, as a more correct measure of this labour for 1 2 hours,
188 kilogrammes raised a kilometre : half the former measure. But
«U this is probably very vague.
17. Among quadrupeds, those which are employed to produce a
mechanical effect are the dog, the ass, the mule, the ox, the camel,
and the horse. Of these the horse is the only one, so far as we are
aware, whose animal energy has been subjected to cautious experi
ments ; and, even with regard to this noble animal, opinions as to
actoal results are very much afloat. The dynamic effort of the horse
is, however, probably about 6 times that of. a strong and active
labourer. Desaguliers states the proportion as 5 to 1, coinciding with
the deductions of Smeaton. The French autliors usually regard
sevep men as equivalent to one horse. As a fair mean between
theae, I assumed in vol. i. of my Mechanics the proportion of 6 to 1,
and stated the strength of a horse as equivalent to 420 lbs. at a dead
pull. But the proportion must not be regarded as constant, but
obviously varies much according to the breed and training of the
animal, as well as according to the nature of the work about which
he is employed. Thus the worst way* as De la Hire observed, of
860 ANIMAL STRBNOTH: H0R8B8 [PABT II.
applying the strength of a horse is to mal«c him carry a weight up a
steep hill ; while the organization of a man fits him very well for tiiat
kind of lahour: hence three men, climhing up such a hill with a
weight of 100 Ihs. each, will proceed faster than a horse with a load
of 300 Ihs.
18. In the memoirs of the French Academy for 1703 are inserted
the comparatiye ohservations of M. Amontons, on the velocity of
men and of horses ; in which he states the velocity of a horse loaded
with a man and walking to he rather more than 5 feet per second,
or 3 1 miles per hour, and when going a moderate trot with the same
weight to he ahout 8J feet per second, or ahout 6 miles per hour.
These velocities, however, are somewhat less than what might have
heen taken for the mean velocities.
19. But the best way of applying the strength of horses is to
make them draw weights in carriages, &c. To this kind of labour,
therefore, the inquiries of experimentalists should be directed. A
horse put into harness, and making an effort to draw, bends himself
forward, inclines his legs, and brings his breast nearer to the earth ;
and this so much the more as the effort is the more considerable.
So that, when a horse is employed in drawing, his effort will depend,
in some measure, both upon his own weight and that which he
carries on his back.
Indeed it is highly useful to load the back of a drawing horse to a
certain extent; though this, on a slight consideration, might be thought
to augment unnecessarily the fatigue of the animal : but it must be
recollected that the mass with which the horse is charged vertically
is added in part to the effort which he makes in the direction if
traction, and thus dispenses with the necessity of his inclining so
much forward as he must otherwise do : and may, therefore, under
this point of view, relieve the draught more than to compensate for
the additional fatigue occasioned by the vertical pressure. Carmen,
and waggoners in general, are well aware of this, and are commonly
very careful to dispose of the load in such a manner that the shaf)^
shall throw a due proportion of the weight on the back of the shall
horse. This is most efficaciously accomplished at Yarmouth, in Nor
folk, where a number of narrow streets connecting the marketplace
i^dth the quay, have led to the invention and use of the low, strong,
narrow carts, thence denominated Yarmouth carts, drawn by one
horse; and on which the loads are frequently shifted, especially when
the vehicles pass over the bridge, in order to give the animals better
foothold, and consequently a greater dynamic effort. •
20. The best disposition of the traces during the time a horse is
drawing is perpendicular to the position of the collar upon his breast
and shoulders : when the horse stands at ease, this position of the
traces is rather inclined upwards from the direction of the road ; bat
when he leans forward to draw the load, the traces should then be
come nearly parallel to the plane over which the carriage is to be
drawn ; or, if he he employed in drawing a sledge, or any thing wiili
• out wheels, the inclination of the traces to the road should (from the
CHAP. Vn.'] ANIMAL 8TRKNGTH : HOftSIS. 361
table at page 52QO) be about 18% when the friction is onethird of
die presfiure. If tlie relation of the friction to the pressure he dif
ferent from this, the same tahle will exhibit the angle which the traces
most make with the road.
21. When a horao is made to more in a circular path, as is often
pnctised in mills and other machines moved by horses, it will bo
necesMry to give the circles which the animal has to ^'alk round the
greatest diameter that will comport with the local and other con
diuuns to which the motion roust be subjected. It is obvious, indeed,
tLat, since a rectilinear motion is the most easy for the horse, the less
the line in which he moves is curved, with the greater facility he will
walk over it, and the less he need recline from a vertical )osition :
and besides this, with equal velocity the centrifugal force will be less
b tbe greatest circle, which will proportionally diminish the friction
of tbe cylindrical part of the trunnions, and the labour of moving the
machbe. And, further, the greater the diameter of the horse walk,
tbe nearer the chord of the circle in which the horse draws is to coin
cidence with the tangent, which is the most advantageous position of
tbe line of traction. On these accounts it is that, although a horse
man draw in a circular walk of 18 feet diameter, yet in general it is
advisable that the diameter of such a walk should not be less than
25 or SO feet ; and in many instances 40 feet would be preferable to
either.
ii. It has been stated by Desaguliers and some others, that a
borse employed daily in drawing nearly horizontally can move, during
cigbt hours in the day, about 5iOO lbs. at the rate of 2i miles per
hoar, or 3 feet per second. If the weight be augmented to about
240 or 250 lbs., the horse cannot work more than six hours a day,
and that with a less velocity. And, in both cases, if he carry some
weight, he will draw better than if he carried none. M. Sauveur
estimates the mean effort of a horse at 175 French, or 189 avoirdupois
pounds, with a velocity of rather more than three feet per second.
But all these are probably too high to be continued for eight hours,
&T after day. In another place Desaguliers states the mean work of
a horse as equivalent to the raising a hogshead full of water (or 550
lbs.) 50 feet high in a minute. But Mr. Smeaton, to whose authority
much is due, asserts, from a number of experiment*, that the greatest
effect is the raising 550 lbs. forty feet high in a minute. And, from
wne experiments made by the Society for the Encouragement of
Arts, under the direction of Mr. Samuel Moore, it was concluded,
that a horse moving at the rate of three miles an hour can exert a
force of 80 lbs. Unluckily, we are not sufficiently acquainted with
tlw nature of the experiments and observations from which these
MnctioDs were made to institute an accurate com]iarison of their
'«*Qlt». Neither of them ought to express what a horse can draw
"pon a carriage ; because in that case friction only is to l)e overcome
(«fter the load is once put into motion); so that a middling horse, well
applied to a cart, will often draw nmcli more than 1000 lbs. The
^^^ estimate would be that which measures the weight that a
3G2 ANIMAL STBSKGTH: H0B8B8. [PABT n.
horse wonld draw up out of a well ; the animal acting hj a horizontal
line of traction turned into the yertical direction hj a simple pulley,
or roller, whose friction should he reduced as modi as poesihle.
23. Mr. Tredgold, in his valuahle puhlication on Railroads, has
directed his attention to the suhject of *'*'hor9e power" The follow
ing is his expression for the power of a horse, 250 ^ \l ) ; and
\ V / for the day's work in Ihs. nused one mile ; d
1 f «
heing the hours which the horse works in a day, and the weight of
14*7
the carriage to that of the load as n : 1. He also gives r for the
a/ d
greatest speed in miles per hour, when the horse is unloaded. These
expressions must, at present, he regarded as tentative. The follow
ing is his tahlc of the comparison of the duration of a horse's daily
labour and maximum of velocity, imloaded.
Duration of labour. Max. Telocity unloaded
Hours, in mile* per hour.
1 147
2 104
3 85
4 73
5 66
6 60
7 65
8 52
9 49
10 46
Taking the hours of labour at 6 per diem, the utmost that Mr
Tredgold would recommend, the maximum of useful effect he assigns
at 125 lbs. moving at the rate of three miles per hour, and regardiiif
the expense of carriage, in that cose, as unity ; then —
Miles per hour. Proportional expense. Moving fbroe.
2 li or M25 166 lbs.
3 1 125
3J 1^7 or 10285 104
4 li or 1125 83
4j l or 1333 62j
5 l or 18 41$
5i 2 36J
That is, the expense of conveying goods at 3 miles per hour bang
1 ; the expense of 4 miles per hour will be 1^ ; and so on, the eX'
pense being doubled when the speed is 5^ miles per hour.
24. Thus, according to Mr. Tredgdd^ wc have for the day of
CfiAK TU.] ANIMAL 8TB1N0TH *. HOBSES. 363
6 hours 2d60 lbs. raised one mile. And Mr. Bevan^ who has made
many experiments on the force of traction to move canal boats on the
Grand Junction Canal, found the force of traction 80 lbs., and the
space travelled in a day 26 miles ; hence, it is only equivalent to
26 X 80 = 2080 lbs. raised one mile for the day's work ; the rate
of travelling being 2*45 miles per hour ; and the result a little less
than Mr. Tredgdd'Sy the difference probably arising from the devia
tion of the angle of the catenary formed by the rope from the horizon.
25. The following experimental data from Mr. Bevan also deserve
attention.
** In the period from 1803 to 1809, I had the opportunity of as
certaining correctly the mean force exerted by good horses in draw
ing a plough ; having had the superintendence of the experiments on
that bead at the various ploughing matches both at Woburn and Ash
ridge, nnder the patronage of the Duke of Bedford and the Earl of
Bridgewater. I find among my memoranda the result of eight
ploughing matches, at which there were seldom fewer than seven
teams as competitors for the various prizes.
lbs.
The first result is from the mean force of each horse in six
teams, of two horses each team, upon light sandy soil = 156
The second result is from seven teams of two horses each
team, upon loamy ground, near Great Berkhampstcad... = 154
The third result is from six teams of four horses each team,
with old Hertford^ire ploughs =: 127
The fourth result is from seven teams of four horses each
team, upon strong stony land (improved ploughs) = 167
The fifth result is from seven teams of four horses each
team, upon strong stony land (old Hertfordshire ploughs) := 103
The sixth result is from seven teams of two horses each
team, upon light loam = 177
The seventh result is from five teams of two horses each,
upon light, sandy land = 170
The eighth result is from seven teams of two horses each
team, upon sandy land = 160
" The mean force exerted by each horse from fiftytwo teams, or
one hundred and fortyfour horses, equals 163 pounds each horse, and
although the speed was not particularly entered, it could not be less
than the rate of two miles and a half an hour.
*^ As these experiments were fairly made, and by horses of the
common breed used by farmers, and upon ploughs of various coun
Uea, these numbers may be considered as a pretty accurate measure
of the force actually exerted by horses at plough, and which they are
able to do without injury for many weeks; but it should be remem
bered, that if these horses had been put out of their usual walking
pace, the result would have been very different. The mean power
of the draughthorse, deduced from the abovementioned experi
ments, exceeds the calculated power from the highest formula of
364
canals: railboaos.
[pari II.
Mr. Leslie^ which is as follows : (15 — o)^ = lbs. aToirdnpois for
the traction of a strong horse, and (12 — v)* ^= lbs. traction of the
ordinary horse, v = velocity in miles per hour."
TABLE l. — Shomtig the maximum quantity of labour a Hone of
averctye strength is capable of performing^ cU different velocities^ on
Canals^ Railways, and Tumptkeroads. {From Tredgold.)
Useful eflfect of one horse
workinir one
Duration
day, in tons drawn one mile." I
Velocity In
of the day'*
work at the
1
miles per
traction In
hour.
preceding
lbs.
On a lerel
velocity.
On a canal.
railway.
level turn
pIkeroML
milc«.
houn.
lbs.
tons.
tons.
tons.
^
u\
83i
520
115
14
3
8
83i
243
92
12
3J
4
83i
153
82
10
4
83i
102
72
9
5
a/o
83i
52
57
72
6
2
83J
30
48
60
7
U
83i
19
41
51
8
n
83^
128
36
45
9
^?.
83i
90
32
40
10
H
83i
66
288
36
Where horse power is employed for the higher velocities, the ani
mals ought to be allowed to acquire the speed as gradually as pos
sible at the first starting. This simple expedient will save the pro
prietors of horses much more than they are aware of; and it deserves
their attention to consider the best mode of feeding and training
horses for performing the work with the least injury to their animiu
powers.
To compare the preceding table with practice at the higher velo
cities, it will be necessary to have the total mass moved, which is
onethird more than the useful effect in this table. Now, the actual
rate at which some of the quick coaches travel, is 10 miles an hour;
the stages average about 9 miles ; and a coach with its load of lug
gage and passengers amounts to about 3 tons ; therefore the average
day's work of 4 coach horses is 27 tons drawn one mile, or 6 tons
drawn one mile by one horse. The table gives 3*6 tons, addea } of
3*6 = 4*8 tons drawn one mile for the extreme quantity of labour
for a horse at that speed, upon a good level road ; from which should
be deducted the loss of effect in ascending hills, heavy roads, &C.,
which will make the actual labour performed by a coachhorse ave
rage about double the maximum given by the table. The conse
quences are well known.
Accordinjr to Mr. Bevans observations, the horses on the Grand
CftAP. VII.] TRACTION ON CANALS, RAILROADS. 365
Jonotion Canal draw 617 tons one mile, at the velocity of 2*45 miles
per hour.
According to Mr. TrtdgM^ if V he the maximum velocity of a
horse, and id any other velocity, the immediate power of a horse is
250 r f I " T7 ) ; and, when the weight of the vessel or carriage
is to the weight of the load, as n: 1, we have V V / =:
1 Hn
the effective power ; and d being the hours the horse works in one
day, the day's work will be V VV in lbs. raised 1 mile,
1 +n
and 250 f I ^ 1 J = the force of traction in lbs. But if the force
14*7
were immediately applied, the value of V would be —7); and to
find the value when the waggons alone are moved, we have
1 147 147
1 : / , : : — , : ~" 7~:., — ; = V ; whence the day's work
v/l + « s/ d y/d (1 + n) ' ^
, fl60dv / v^d(\^n)\ .... . ,
18 I 1 —\)= ' ] I which IS a maximum when
1 + n \ 147 /'
06
= d. Consequently, when the velocity is given, we
r* (1 + «)
06
bftve 77 r equal the duration of the day's work in hours:
tr(l f 11) ^
8000 / 0*8 \
and — rr Tj = the effective day's work; and 250 f I — — j =
83^ lbs. But we may assume n to be always so near ^, as not to
affect the result : and then, — = rf, and = the day's work in
2
lbs. or very nearly  tons raised one mile. This, being combined
with the Dumbers of the preceding table, gives the effect of a horse
on canals, railroads, and turnpikeroads.
It must, however, be here added, that although the deductions
from Mr. Tredgdd^% valuable tables, as to the effects on canak^ are
tolerably accurate up to rates of 4 or 5 miles per hour, yet, when
boats are moved on canals at rates of from 9 to 12 or 14 miles
per hour, the circumstances of the resistances undergo an essential
ehaoge. The resistance, in fact, becomes so small, that passage
366 TaACTION ON CANALS, RAILBOADS. [PART II.
boats DOW trayel at these high velocities; and it is hence probable
that railroads and canals will admit of a competition such as the sup
porters of railroads never anticipated.
I shall here briefly detail some of the facts, as they have been
given in a letter widely circulated by Mr. W. Grakame of Glasgow,
in the Nautical Magaziney and other places.
From the traffic by canal boats, which has been actually going on
during the last two years and a half, on the Paisley canal, we learn
this remarkable fact, that, while a speed of ten miles per hour has
been maintained by the canal boats, the banks have sustained no in
jury whatever. The cause of injury, in truth, has been entirely sup
pressed by the velocity of the boat, which passes along the water
without raising a ripple.
About two years ago, measures were adopted for increasing the
speed of the boats on the Paisley or Ardrossan canal. This canal is
by no means favourable to such experiments, being both serpentine
in its course, and narrow : it connects the town of Paisley with the
city of Glasgow, and the village of Johnstone ; the distance being
about twelve miles. The boats employed on this canal are 70 feet
in length, and 5*6 broad, and carry, if necessary, upwards of 120
passengers. They are formed of light iron plates, and ribs covered
with wood, and light oiled cloth, at a whole cost of about 1 25/. They
perform stages oJT four miles in an interval of time varying from 22
to 25 minutes, including all stoppages, and the horses run three or
four of these stages alternately every day. The passengers are under
cover, or not, as they please, no difference being made in this parti
cular ; and the fare is one penny per mile in the first, and three
farthings per mile in the second cabin.
The horses drawing the canal boat are guided by a boy, who rides
one of them ; and, in passing under bridges at night, a light is shown
in the bow of the boat, by which he sees his way, and which light
is closed when the bridge is passed. Intermediate passengers are
also accommodated to distauces even as small as a mile ; and the faci
lity with which the boat stops, when reHeved from the drawing force,
is such as avoids all danger whatever. The expense of conveying a
load of eight tons at a rate of nine or ten miles per hour^ including
all outlay, interest, and replacement of capital, is not more than i\d.
per mile. It is also ascertained that one ton weight may be carried
on a canal at nearly the same speed as on the railway, at about 1^
per mile, including an allowance for interest and replacement of
capital.
It is also believed, that if the breadth and curvature of the Paisley
canal admitted boats of 90 feet length, instead of 70, they would
carry more passengers by onehalf without an additional expense, and
a decrease of labour to the horses.
The foregoing has been deduced from calculations founded oo
the observation of facts relating to the wear and tear of boats and
horses, and the absolute resistance which these boats meet with in
passing through the water. On this subject it has been observed.
CHAP. VU.] TRACTION ON CANALS, RAILROADS. 3^7
that, in addition to the common resistance of the water to the
motion of the boat, a wave, or body of water, is also raised before
it, varying in its height according to the velocity of the boat, and
constantly presenting an obstacle to ber progress, providing that she
only moves through the water at a certain slow rate. The height of
this wave will then amount to nearly two feet, often overflowing the
banks of the canal, and, from the obstruction it occasions, eventually
obliging the boat to be stopped. Now, if, instead of stopping the
boat when this wave is raised, her velocity be increased beyond what
it had then been, she advances and passes over it, and leaves it to
subside in her wtike, which it does, and the water becomes perfectly
still. The same horses, drawing the boat at this increased speed, are
found to perform their work better, the resistance to their progress
having become less ; and the more the velocity of the boat is thus
increased, the less resistance she meets with, merely having to cut
the still water instead of the wave. It is a curious fact, that the
wave produced by the approach of a slow canal boat is often observed
at the distance of a mile, and upwards, along the canal, before the
arrival there of the boat. But, in the case of the high wave being
raised by the Paisley canal boat, it is customary to stop the boat, and
after it has subsided to start again at a greater velocity. When the
boat 18 to be stopped for any purpose, as her speed decreases the
wave rises in proportion, and washes over the banks, until the motion
of the boat becomes so small Iks to produce none. The discovery is
doahtless a very important one, and, if turned to account, is likely to
produce a material alteration in the rate of transport on canals. It
was not known until these experiments were made, that if a boat,
from a state of rest, was dragged along a canal, in proportion as her
•peed increased to a certain limit, that the power required was
greeier: bat that, if she were started at, and preserved a speed ex
ceeding the same limit, the power required would be less, and would
decrease as her velocity increased, in fact, from a certain velocity
there seems to be no limit to the rate at which a boat, as far as ani
mal power can be applied, may thus pass through the water ; and as
the rate increases the power required decreases. On this principle
it is that the boats on the Paisley canal, with ninety passengers in
them^ are drawn by horses at a speed of ten miles an hour; while it
would kill them to draw the same boat along the canal at six mile$
an hour. A boat might indeed travel fifteen or twenty miles an
hour eancr than at six miles. The former of these velocities has
already been attained by Mr. Grahamey along a distance of two miles,
and is considered by him safer both for the boat and tlie canal.
As a proof of what may be done by this method of carriage, Mr.
Chrakame states that he has performed a voyage of fiftysix miles
along two canals in six hours and thirtyeight minutes, which in
cluded the descent of ^ye^ and the ascent of eleven locks, the pas
sage of eighteen drawbridges where the trackinglino was thrown
off, and sixty common bridges, besides a tunnel half a mile long ; all
of course prodacing some delay. The boat which performed this
368 TRACTION UN CANALS AND ROADS. [PART II.
was sixtynine feet long, and nine broad, drawn by two borses, and
carried thirtythree passengers, with their luggage and attendants.
These facts furnish great encouragement to canal companies, to
improve the construction of their boats and the speed on their canals ;
and thus, probably, in some situations, supersede the necessity of
railroads.
Mr. Macneill, the assistant engineer upon the Holyhead road, under
Mr. Telford, in the course of his examination before a committee of
the House of Commons, on steamcarriages, railroads, &c. gave the
subjoined curious information.
Wellmade roads, formed of clean hard broken stone, placed on a
solid foundation, are little affected by changes of atmosphere ; but
weak roads, or such as are imperfectly formed i^dth gravel, flint, or
round pebbles, without a bottoming of stone, pavement, or concrete,
are much affected.
On the generality of roads, the proportional injury from the wea
ther and traffic is nearly as follows. When travelled by fast coaches :
from atmospheric changes 20 ; coachwheels 20 ; horses' feet 60 =
1 00. When travelled by waggons : atmospheric changes 20 ; waggon
wheels 35'5 ; horses' feet 44*5 = 100, Has ascertained, from a num
ber of observations, that the wear of the iron tire of fastgoing coach
wheels is, compared with that of the shoes of the horses which draw
them, as 326*8 to 1000, or as 1 to 34ths nearly ; and infers that the
comparative injury done by them to roads is nearly in the same pro
portion. In the case of slowgoing carriages and horses the propor
tion is as 309 to 360, or as 1 to 1*16, or nearly 1 to 1. The tire of
the wheels of the fastgoing coaches last from two to three months,
according to the weather, the workmanship, and quality of iron;
about 20 years ago, it did not lost seven days on an average. Coach
horse shoes remain in use about thirty doys; waggonhorse shoes
about five weeks on an average. Where roads are weak, and yield
under pressure, the injury caused by heavy wheels is far greater than
on solid firm roads.
It was found, in one instance, that the wear of haid stone, placed
on a wet clay bottom, was four inches, while it was not more
than half an inch when placed on a solid dry foundation. On the
Highgate archway road, the annual wear is not more than half an
inch in depth. To the same gentleman we owe the following useful
table.
TABLE II. — Tlie general Result of Experiments made with a Stage
Coachy weighing^ exdtuive of seven passengers^ 18 ctct.^ on the same
piece of road^ on different inclinations^ and at different rtUes of
velocitgy furnish the following statement.
Rate of Inclination. Rates of Travelling. Force required.
I in 20 6 miles per hour 268 lbs.
1 in 26 6 213
1 in 30 6 165
CHAX».
VII.]
Hate of Inclination.
1
TRACTION ON ROADS.
369
Rates of Travelling. Force required.
n 40 6 miles per hour 160 lbs.
In 600 6 Ill
20
26
30
40
in 600 8
20
26
30
40
10
10
10
10
n600 10
296
219
196
166
120
318
225
200
172
128
B B
370 STBEKOTH OF MATERIALS. [PART II
CHAP. VIII.
STRENGTH OP MATERIALS.
Sect. I. EestdU of Experiments^ and Principles upon which tkjf
should he practically applied.
By most writers on the strength of materials, it has been customarj
to start with the assumption of the three undermentioned principles,
and upon them to construct a theory by means of which they afte^
wards deduced from experiments on rectangular bars tlie strength of
beams of the same material, but of various forms of transTerse sec
tion. Upon, however, testing these deductions by experiment, thcj
are found in many cases to be entirely erroneous, and to such an
extent as to render their practical application not only uncertain, bat
frequently highly dangerous. Such being the result of experience,
it becomes important to examine these principles more closely, in
order to discover wherein the cause of this erroneous result consists.
The principles which we allude to are (in the language of Tred
gold*) as follows: —
" The first is, that the strength of a bar or rod to resist a given
strain, when drawn in the direction of its length, is directly pro
portional to the area of its cross section ; tchile its Mastic pc^n^
remains perfect^ and the direction of the force coincides wiUi the
axis.
" The second is, that the extension of a bar or rod by a fore*
acting in the direction of its length is directly proporUonal to tbe
straining force, when the area of the section is the same ; whiU ^
strain does not exceed the elastic power.
" The third is, that while the force is within the elastic power of^
material^ bodies resist extension and compression with equal forces.
It should be remarked, that each of the foregoing propositions i«
only asserted to hold good " while the strain does not exceed the
elastic power of the material." By which is meant that tbe force
applied shall not be sufficient to cause any permanent displacement of
the particles, and that when the straining force is removed, tbe body
will not have acquired any set, but will reassume its former sbspc
* Practical Treatise on the strength of cast iron. Fourth edition, pw ISi
STRSNOTH OF MATBKIALS. 371
». Tredgold and most other writers upoD this subject
d that no permanent displacement or set takes place
ining force amounts to about onethird of tho ultimate
be material, and have consequently assumed that these
ght be applied without error to determine the strength
sv of materia], of any form, and subject to any kind
more careful experiments have, however, shown conclu
permanent displacement or set takes place with a much
I than onethird of the ultimate strength, and that in
able set is produced by a straining force very much less
which the material will be likely to be exposed in
cfa being the case, it is obvious that these principles (as
»Te) are inapplicable to any practical case. As, how
ire been disposed to imagine that such a strict limitation
icmtion is not necessary, and that they ])ractically hold
ben the force applied eaxeeds the elastic power of the
ill be desirable to show that not only is such a supposi
nroneous, but that the principles themselves, even with
t, so far from being demonstrable, are, on the contrary,
loubtful.
■ertion, that the strength of a rod or bar subjected to
n is directly proportional to the area of its cross section,
irat sight it may appear obvious enough, nevertheless is
practice to be strictly correct ; for, if the material is of
it found that the strength of the exterior hard crust is
bat of the interior substance, and therefore that the
bicb the perimeter of the section bears to its area is a
requiring to be taken into consideration ; not to men
urangement of the particles during the cooling of the
ipon which arrangement the strength of the substance so
lends, is greatly influenced by the external form of its
i on the other hand, in the case of a fibrous material, aa
da or timber, in consequence of the fibres not laying
illel to each other, in cutting the bar to the required I
may of the exterior fibres will be cut transversely, and >
rfbre be capable of affording so great a proportionate li
m similar fibres within the more central portion of the jj
ieh it follows, that the tensile strength of a bar of any
i directly proportional to its sectional area, although tho I
» trifling as not to require attention in most of the cases ij
i tfaemselyes in practice. {
I law, that the extension of a bar or rod exposed to a \^
m directly proportional to that strain 90 long cu ike area \
remains unaltered^ is from the very circumstances of \
■ible, since it is not capable for a bar to become elon
biTing its transverse section proportionately diminished,
we cannot conceive the bar in two successive degrees I
l^Ting in both the same sectional area. And although
B B 2
372 STRENGTH OF MATBBIALS. [PABT IL
the correct law subsisting between the elongation and the force pro
ducing it, has not yet been satisfactorily determined, there is eierr
reason to believe that they are not directly proponionil to em
other.
With regard to the third law, which may be more definitely ititfld
as follows, viz. : — that within the elastic power of the materaJ,bo&i
require as great a force to alter their dimensions to any giTen exttnt
by compression as by extension ; we may remark, that an? propeilf
depending upon the straining force being within the elastie power tf
the material is incapable of any practical application, and tbeRfim
that this law, as limited above, even assuming it to be correet, doM
not require to be included in any practical inquiry into the kvt
which regulate the strength of materials. And without soch liniti^
tion the law is at utter variance with the results of experiment, net
no fact is now better ascertained than that cast iron requires a dmIi
greater force to compress it to any given extent, than to extend it to
the same extent.
Such then being the result of our investigation into the socoiteeB
of these principles with practical experience, we cao no longer It
surprised that the deductions drawn from them by Tredgc^ mI
others, should be equally at variance with the same experieaefc
Indeed the errors into which they have fallen appear to hate beet
caused by reasoning upon hypothetical principles — attempthig to foia
a complete theory of the strength of materials upon data neitlMr
sufficient in quantity nor certainty. And even now, although una
has been done, since the time when Tredgold wrote, by HodgkiMi
Fairbaim, Thomas Cubitt, Barlow, and some others, this inraffieienef
has not yet been supplied, and we are still in ignorance upon OMJ
points, without a knowledge of which it would be impossible to fi»«
any complete theory of the strength of materials aafiicient in Hi
extent for all practical purposes, and of whose accuracy we cooM he
so far sure as to allow of its application without fear of erroseov
results.
In order to show in what points the deficiency in oor knowle^
upon this subject consists, and to explain the general prindplei vp**
which the strength of materials actually depends, we will hiiiif
investigate the subject
If A BCD, fig. 230, be a rectangular beam, subjected to fcj
pressures Pj, P.^, and P.,, the two former of which arc eqoal to eidi
other, and to half P,, and if Pi and P, be applied at the two eAUHJi
ties of the beam, and P.< at its center in an opposite direetioBi ■•
beam will be deflected, the upper surface A C becoming cooviX» •*
the lower surface BD concave. Then, if the material compoiiogvt
beam be supposed capable both of extension and com prcMi o^ w*
fibres on the convex side will be extended, and those on the con**
side compressed ; there will, however, be a certain curved aupeifc*
£ F within the beam (called the neutral superficies), the fibres tfti^
in which will not be either extended or compressed.
Now, if tangents EG, FU, to this curved neutral miperficiei it ^
CHAP. VIII.] STRENGTH OP MATEBIAL8. 373
extremities of the beam be drawn, and from them perpendiculars K L,
M N, pasnng through the point I, in which the neutral superficies inter
lects the plane of rupture OQ; then will the triangle NIL be equal to
the sum of the extensions of the whole length of all the extended fibres;
tnd the similar triangle KIM will be equal to the sum of the com
pressions of the whole length of all the compressed fibres; and if any
•operfides be taken between AC or BD and the neutral superficies, the
portions of such superficies, as t^ i,, yi^s? contained between KL, and
II N, will be equal to the extension or compression of the whole length
of the fibres situate in that superficies : from whence it follows that the
amount of extension or compression of any fibre is directly propor
tional to its distance from I, the neutral axis.
If now any number of points be taken in the depth of the beam,
as ij, f^ 1^ yj, ^29 y»* ^^' (%• 231), and lines be drawn through
them perpendicular to O Q, each equal in length to the force required
to extend or compress the fibres to the extent to which such fibres
are actually extended or compressed, and which extent will be propor
tiona] to their distances from the neutral axis, and a curved superficies
^^i^iCtdid^e^l^bj^a^ be drawn through the ends of all these lines,
then will the solid contained between the plane of rupture O Oj Q^ Q
and the curved superficies, be equal to the resistance of all the extended
and compressed fibres, or to the transverse strength of the beam.
Now, if £ equals the solidity of ai^^lil^^^^a) ^^ ^ equals the
soKdifcy of ^iCid^^d^c^I^ also ^and A equal the perpendicular distances
of their center of gravity from the neutral superficies, and ^L the dis
tance of the points where the pressures P^ and F^ are applied, from the
neutral axis ; we have from the principle of the equality of moments,
iLP^ = E^ + CA;
or, ainoe El must be equal to C^, and P^ equals \ P,, we have
5l'P,= E» (I.)
■ad this formula will apply» whatever form the transverse section of
the beam may be, whether rectangular as fig, 281, cylindrical as
fig. 232, or X shaped as fig. 233.
It is further evident £at, when the beam is upon the point of
breaking, the upper fibres at O must be strained to the utmost extent
Id whkh they can be, before yielding, and that as this extent does
not depend upon the form of the beam, but only upon the elastic
properties of tne material, for the satee material the length a^ O will
renuitn constant in all cases. From this it follows that the lines
■i, f^ •„ 7„ y^ 73, &c. (fig. 231), which are the ordinates of the
enire a^hA^e^di^ will also be constant, but that their distances from
E, or the Miacisse of the curve will vary directly as the distance O I
of the neutral axis from the upper surface of the beam.
In rectangular beams, or others, whose transverse sections are
■mikr figures, the distance O I, and therefore the area of the curve
374 STRBNOTH OP MATERIALS. [PART II.
O a^ b^ Ip will yary as the depth of the beam, and the solidity of the figure
Oa^b^ IjI^^^.a^O will vary as the depth and the breadth, or Eo: bd;
also, since the forms of the beams are similar^ the distance i of the
center of gravity of the solid E from the neutral superficies, mnst vary
as the depth of the beam ; and therefore, in rectangular beams, the
ultimate strength of the beam varies directly as the square of the depth
multiplied by the breadth, and inversely as the length ; or, putting
S (= Pj H Pj rs P.^) for the breaking weight applied in the center,^
we have
Sac^* (II.)
or, when the transverse sections are similar figures^ </^ being any
similar dimensions in both,
d^
Sa^ (III.)
From the preceding investigation it is evident that we cannot
deduce any general formulse by which to determine the transverse
strength of a beam of any given form of section, without knowing
the position of the neutral axis, the limit to which the fibres may
be extended before yielding, and the law which subsists between the
extension and compression of the material and the force required to
produce such extension and compression ; upon none of which sub
jects have we any certain knowledge.
Sect. IL Stren^ ofMateriaU to resist tensile and crushing strains*
Strength of Columns.
The first principle alluded to in the foregoing section, via., that the
strength of a bar or rod to resist a tensile strain in the direction of
its axis, IS proportional to the area of its cross section, is not, for the
reasons there mentioned, strictly correct. And although perhaps for
ordinary practical purposes, or where the sections to be compared aie
large, no error of importance wonld result from the adoption of this
principle, it is very necessary, in experiments whose object is tbe
determination of the actual strength of any given materia], that doe
regard should be had to all circumstances, (however trivial they msj
seem,) which may in any way affect the accuracy of the results.
The only late experiments upon the tensile strength of cast iron,
now before the public, are those of Mr. Eaton Hodgkinson, first
published in a paper communicated to the British Association, and
subsequently in his '^ Experimental Researches." The mean tensile
strength of a square inch of cast iron, as deduced from these experi
ments, is only 10,560 lbs., or 7 tons 7*85 cwt.; the lowest heing
12,993 lbs., and the Iiighest 21,907 lbs. In a series of experiments,
OHAP. VIII.] C0HE8IVB 8TRINOTH OF CAST IRON. 375
howeTer, which have been tried by Mr. Thomas Oubitt*, during a
period of several years, we find the strength of cast iron to be con
siderably higher than Mr. Hodgkinson's estimate: the average of
twentyfive specimens, of which eight were unsound, gave 27,140 lbs.
for the tensile strength per square inch section ; and, exchiding the
unsound pieces, the average of the others gave 27,773 lbs., or 12 tons
8 cwt. The highest broke with 32,997 lbs. per square inch, and the
lowest^ even of the unsound pieces, broke with 21,471 lbs., or only
a little under Mr. Hodgkinson's best speeimen.
The anomaly between these results is very remarkable, since in
both cases the experiments were conducted with the greatest care ;
we think, however, that Mr. Cubitt's results are entitled to the pre
ference, for this reason, that while any cause might easily operate to
lessen ihe strength of the iron, we cannot conceive it possible that
any want of care in the experiments or bad arrangement of the
apparatus could cause the iron to bear more than it otherwise would
have done, providing that the means of measuring the force exerted
in producing fracture were correct ; and these, we do not hesitate to
aay, were not open to any exceptions. It is certainly difficult to
account for so wide a difference in the results of these experiments,
but we cannot but think that the form chosen by Mr. Hodgkinson,
for the transverse section of the bars upon which he experimented, is
open to objection. The form adopted by him is shown in fig. 234,
being that of a cross ; and the reason which induced him to choose
this form, was " to obviate the objection made by Mr. Tredgold
(Art. 79 and 80,) and others, against the conclusions of former
experimenters." The objection of Mr. Tredgold here alluded to was,
that if the strain did not pass through the center of gravity of the
section in the place where the rod broke, the different parts of that
section would be exposed to a different strain, and therefore,' that the
results obtained would be lower than the truth. We quite agree with
Mr. Tredffold upon the importance of this objection, but we cannot
bat consider, that the form chosen by Mr. Hodgkinson is more open
to this objection than the square or cylindrical form would have been.
For, whatever might be the form of section of the rod, the same means
of fixing the ends of it might be used, and since it depends entirely
upon the accuracy with which that is done whether the axis of the
rod coincides with the straight line between the points of attachment,
(which is the direction of the straining force), we may suppose that
the lateral distance between these two lines at the breaking section
would be the same, whatever was the form of that section; such
* I am much indebted to Mr. Thomas Cubitt, for the liberal manner in
wliich he has placed at my disposal the results of his very careful and ela
bormte experiments upon this subject. Ui« experiments on the set, deflection,
and ultimate strength of cast iron girders, extend over a period of many years,
and p o i Bs eis the immense practical advantage of having been made upon large
girders. These experiments, amounting in number to upwards of 2000, have
been taken under the direction of Mr. Dines; and, having witne98ed some of
them myself, I can bear testimony to the care which has been taken to ensure
aecurmte results, and to remove all imaginable causes of error.— H. L.
370 C0HE8IVB 8TRBN0TH OP MATERIALS. [PART II.
being the case then, we hold that the strength of the cnicifonn sec
tion would be more affected than the square, since in the former the
metal is situated further from what would then be the neutral axis
of the section, and exposed therefore to greater differences of strain.
This form is also open to the practical objection that it is very diffi
cult to obtain a perfectly sound casting from it ; it asually happens
that the iron is not perfect at the point of intersection, (a, fig. 234,)
and also to our remarks at p. 371, that the proportion of the peri
phery to the area may influence the strength.
Mr. Cubitt has also tried a few experiments upon wrought iron,
which gave for the average tensile strength per square inch,
58,952 lbs., or 26 tons 6*3 cwt.
The metals differ more widely from each other, in their elastic
force and cohesive strength, than the several species of wood or vege
table fibres. Thus, the cohesion of fine steel is about 135,000 lbs.
for the square inch, while that of cast lead amounts only to about the
hundred and thirtieth part, or 1800 lbs.
According to the experiments of Mr. George Rennie, in 1817, the
cohesive power of a rod an inch square of different metals, in pounds
avoirdupois, with the corresponding length of the modulus of elasticity
in feet, is as follows : —
Pounds. Feet.
Cast steel 134,256 ... 39,455
Swedish malleable iron ... 72,064 ... 19,740
English ditto 55,872 ... 16,938
Cast iron 19,096 ... 6,110
Cast copper 19,072 ... 5,003
Yellow brass 17,958 ... 5,180
Cast tin 4,736 ... 1,496
•Cast lead 1,824 ... 348
It thus appears, as Professor Leslie remarks, that a vertical rod o£
lead 348 feet long would be rent asunder by its own weight. Ther*
best steel has nearly twice the strength of English soft iron, and this
again is about three times stronger than cast iron. Copper and brass
have almost the same cohesion as cast iron. This tenacity is some—
times considerably augmented by hammering or wiredrawing, that=
of copper being thus nearly doubled, and that of lead, according t o^^
Eytelwein, more than quadrupled. The consolidation is produced ^
chiefly at the surface, and hence a slight notch with a file will
materially weaken a hard metallic rod.
Professor Leslie has given the following tabular view of the
lute cohesion of the principal kinds of timber, or the load which^
would rend a prism of an inch square of each ; and the altitude of tb^?
prism which would be severed by the action of its own weight.
Pounds. Feet.
Teak 12,915 ... 36,049
Oak 11,880 ... d2,900
Sycamore 9,630 ... 35,800
Beech 12,225 ... 38,940
CHAP. VIII.] STRENGTH OF CORDAGE. 377
Pounds. Feet.
Ash 14,130 ... 42,080
Elm 9,720 ... 39,050
Memelfir 9,540 ... 40,500
Christiana deal 12,34G ... 55,500
Larch 12.240 ... 42,160
The following is the result of Professor Barlow's experiments upon
the cohesive strength of various kinds of timber.
Pounds
per square inch.
Fir 12,203
Ash 17,077
Beech 11,467
Oak 10,389
Teak 15.090
Box 19,891
Pear 9,822
Mahogany 8,041
The cohesion of hempen fibres is, for every square inch of their
transverse section, 6400 Ihs. The best mode of estimating the strength
of a rope of hemp is to multiply by 200 the square of its number of
inches in girth, and the product will express in pounds the practical
strain it may be safely loaded with ; for cables, multiply by 120 instead
of 200. The ultimate strain is probablv double this, as will appear
from the account following of Du Hamei s experiments. If yams of
180 yards long be worked up into a rope of only 120 yards, it will
lose onefourUi of its strength, the exterior fibres alone resisting
the greatest part of the strain. The register cordage of the late
Captain Huddart exerts nearly the whole force of the strands, since
they suffer a contraction of only the eighth part in the process of
combining.
For the utmost strength that a rope will bear before it breaks, a
good estimate will be found by taking onefi/lh of the square of the
girth of the rope, to express the tons it will carrv. This is about double
the role for practice which we have given above ; and is, even for
an ulterior measure, too great for tarred cordage, which is always
weaker than white.
The following experiments were made by Mons. Du Hamel, at
Rochfort, on cordage of three inches (French) circumference, made
of the best Riga hemp, August 8th, 1741.
White. Tarred.
Broke with a strain of 4500 pounds «.. 3400 pounds.
4000 „ ... 3300 „
4800 „ ... 3258 „
August 25th, 1743.
4600 „ ... 3500 „
5000 „ ... 3400 „
5000 „ ... 3400 „
378 snunroTH op oobdaoi. [part ii.
September 23, 1746.
White. Tmrred.
Broke with a itruo of 3880 pounds ... 3000 pounds.
4000 „ ... 2700 „
4200 „ ... 2800 „
A parcel of wbite and tarred cordage was taken out of a quantity
wLich had been made February 12, 1746.
It was laid up in the Magazine, and comparisons were made from
time to time, as follows : —
Wbite. Tarred. Difference.
1746, April 1 4th, 2645 pounds 2312 pounds 333 pounds.
1747, Mav 18th, 2762 „ 2155 „ 607
1747, October 21st, 2710 „ 2050 „ 660 „
1748, June 19th, 2575 „ 1752 „ 823 „
1748, October 2nd, 2425 „ 1837 „ 588 „
1749, Sept. 25th, 2917 „ 1865 „ 1052 „
M. Du Hamel says, that it is dcnnded by experience, that white
cordage in continued service is onethird more durable than tarred ;
secondly, it retains its force much longer while kept in store ; thirdly,
it resists the ordinary injuries of the weather one fourth longer.
These obsenrations deserre the attention of pmctical men.
Mr. B, Becan has favoured the author with a tabular view of his
results with regard to the modulus of cohesion^ or the length in feet
of any prismatic substance required to break its cohesion, or tear*
it asunder.
Bet>an% Remits,
Feet.
Tanned cow's skin 10,250
Tanned calf skin 5,050
Tanned horse skin 7,000
Tanned cordovan 3,720
Tanned sheep skin 5,600
Un tanned horse skin 8,900
Old harness of thirty years 5,000
Hempen twine 75,000
Catgut, some years old 23,000
Garden matting 27,000
Writingpaper, foolscap 8,000
Brown wrappingpaper, thin 6,700
Bent grass, (holcus) 79,000
Whalebone 14,000
Bricks, (Fenny Stratford) 970
Bricks, (Leighton) 144
Ice 300
Leicestershire slate 7,300
CHAP. VIII.]
8TRBN0TH OF COLUMNS.
879
STRENGTH OF COLUMNS.
Upon the subject of the strength of columns, we are entirely
indebted to Mr. Hodgkinson, who, after a very careful and accurate
series of experiments, has deduced formulee of immense practical
value, and has clearly shown the principles upon which the strength
of materials, submit to a compressing force, depends.
The results of his experiments are briefly as follows : — That when
weights are applied to prisms, or cylinders, the shorter ones usually
bore more than the longer ones, and that the strength may be con
sidered to vary as the area. That, when the diameter or least
lateral dimension is less than the height of the prism, fracture takes
place, by the separation of either a pyramid, cone, or wedee,
(depending upon the form of the prism,) the angle of whose sides
is constant for the same material, and about 34° for cast iron. And
further, that the mode of fracture and the ultimate strength of a
prism varies but slightly with a variation in its height, so long as it is
not less than its diameter or least lateral dimension, or greater than
four or ^ye times the same dimension; when higher, it begins to
bend, and its strength decreases.
Mr. Hodgkinson finds that the strength of a column is very much
influenced by the manner in which the ends arc fixed ; when both
ends are rounded, so that the column may turn upon them as a
center, its strength is only onethird of that of another column of
precisely similar dimensions, but whose ends are flat and incapable
of motion ; and that, if one end is flat and immovable while the other
is rounded, the strength is twothirds of what it would have been had
both ends been flat.
The following table exhibits the results of his experiments on solid
and hollow cylindrical columns of cast iron, in which S is the break
ing weight in tons ; D the external diameter, d the internal diameter,
both in inches ; and / the length or height of the column in feet.
Kind of Cdumn.
With both ends round,! With both endii flat.
when the height of the'when the height of the
column is not less than'oolumn is not less than
15 times iu diameter. 30 times its diameter.
Solid cylindrical cast ironl
columns j
Hollow ditto ditto
S = 149
8 = 13
p3^
/17
8 = 4416
8 = 443
/17
When the height of the column is less than that mentioned in this
table, namely, fifteen times its diameter for columns with both ends
rounded, and thirty times the same with both ends flat, the strength
of the column becomes modified in consequence of its being then
partially crushed as well as bent. If C equal the force which would
be required to erush the column without flexure, 8 the strength as
380 8TRBN0TH OF COLUMNS. [PART II.
calculated by the above formulse for long columns, and S^ the actual
strength, we have
'■JTT' <>^'
The value of c, or the weight required to crush a square inch of
cast iron, as deduced from his experiments, is 1 07,750 lbs., or 48 tons
2 cwt. ; therefore, the crushing force is to the tensile, according to
Mr. Hodgkinson, as G'507 is to 1 ; or, taking Mr. Hodgkinson's value
for the crushing strength, and Mr. Cubitt's for the tensile, the ratio
becomes as 3*88 is to 1.
The strength of a column of cast iron of given dimensions being
1000, the strength of a column of wrought iron of the same dimen
sions would be 1745, of cast steel 2518, of Dantzic oak 108*8, and
of red deal 785.
The weights required to crush cubes of the quarter of an inch of
certain metals, according to the experiments of Mr. Rennie, arc as
follows : —
Pounds.
Iron cast vertically 11,140
Iron cast horizontally 10,110
Cast copper 7,318
Cast tin 966
Cast lead 483
Cubes of an inch are crushed by the weights annexed : —
Pounds.
Elm 1,284
White deal 1,928
English oak 3,860
Craigleith freestone 8,688
Cubes of an inch and a half, and consequently presenting a section
of two and a quarter times greater than the former, might be expected
to resist compression in that ratio. They are crushed, however, with
loads considerably less.
Pounds.
Red brick 1,817
Yellow baked brick 2,254
Fire brick 3,864
Craigleith stone, direction of the strata... 15,560
Ditto across the strata 12,346
White statuary marble 13,632
Whiteveined Italian marble 21 ,783
Purbeck limestone 20,610
Cornish granite ..« 14,302
Peterhead granite 18,636
Aberdeen blue granite 24,536
These facts show the comparative firmness of different materials ;
but it is to be regretted that such results are not of much praetical
value, since they are confined to a very narrow scale, and applicable
CHAP. VIII.] MODULUS OP ELASTICITY. 381
only to cubical blocks. While the breadth remains the same, the
resistance appears to depend on some unascertained ratio of the
altitude of the column.
Sect. III. EUutieiiy and ElongcUion of Bodies suhfected to a
crushing or tensile strain.
The modtdtis of the elasticity of any substance is a column of the
same substance, capable of producing a pressure on its base which is
to the weight causing a certain degree of compression, as the length
of the substance is to the diminution of its length.
The modulus of elasticity is the measure of the elastic force of
any substance.
A practical notion of the moduius of elasticity may be readily
obtained. Let i be the quantity a bar of wood, iron, or other sub
stance, an inch square and a foot in length would be extended or
diminished by the force f; and let / be any other length of a bar of
equal base and like substance ; then
1 : / : : I : ^, or /f = ^,
where A equals the extension or diminution in the length /.
The modulus of elasticity is found by this analogy : as the dimi
nution of the length of any substance is to its length, so is the
force that produced that diminution to the modulus of elasticity. Or,
denoting the weight of the modulus in pounds for a base of an inch
square by m, it is
I :/: : 1 : w = '^.
E
And if v be the weight of a bar of the substance one inch square
and one foot in length ; then, if M be the height of the modulus of
elasticity in feet, we have
M=^ (V.)
The weight of the modulus of the elasticity of a column being
m, a weight bending it in any manner f the distance of the line
of its application from any point of the axis D, and the depth of the
column, dy the radius of curvature will be .
The distance of the point of greatest curvature of a prismatic
beam, from the line of direction of the force, is twice the versed sine
of that arc of the circle of greatest curvature, of which the extremity
is parallel to that of the beam.
When the force is longitudinal, and the curvature inconsiderable,
the form coincides with the harmonic curve, the curvature being pro
portional to the distance from the axis ; and the distance of the point
382 MODULUS OF ELASTICITY. [PABT II.
of indifference from the axis becomes the secant of an arc propor
tional to the distance from the middle of the column.
If a beam is naturally of the form which a prismatic beam would
acquire, if it were slightly bent by a longitudinal force, calling its
depth dy its length /, the circumference of a circle of which the
diameter is unity c, the weight of the modulus of elasticity m, the
natural deviation from the rectilinear form A, and a force applied at
the extremity of the axis /y the total deviation firom the rectilinear
form will be
^^VA^»_
It appears from this formula, that when the other quantities
remain unaltered, t! varies in proportion to A, and if A = O, the
beam cannot be retained in a state of inflection, while the denominator
of the fraction remains a finite quantity; but when d^trm = 12 V f^
A^ becomes infinite, whatever may be the magnitude of A, and the
force virill overpower the beam, or will at least cause it to bend so
much as to derange the operation of the forces concerned. In this
^ j . — , 8225 — w, which is the force capable of
holding the beam in equilibrium in any inconsiderable degree of
curvature. Hence, the modulus being known for any substance, we
may determine at once the weight which a given bar nearly straight
Is capable of supporting. For instance, in fir wood, supposing its
height 10,000,000 feet, a bar an inch square and ten feet long may
begin to bend with the weight of a bar of the same thickness, equal
in length to 8225 x — :: tt^ x 10,000,000 feet, or 671 feet;
120 X 120
that is, with a weight of about 120 lbs. ; neglecting the effect of the
weight of the bar itself. In the same manner the strength of a bar
of any other substance may be determined, either from direct experi
ments on its fiexure, or from the sounds that it produces. If
♦n /* /
/ = — , 'ri = 8225 «, and  = v' (8225 «) = 907 ^ n ; whence,
n a a
if we know the force required to crush a bar or column, we may
calculate what must be the proportion of its length to its depth, in
order that it may begin to bend rather than be crushed.
When a longitudinal force is applied to the extremities of a straight
prismatic beam, at the distance D from the axis, the deflection of the
middle of the beam will be
"•{•(V^^)^} ^"""^
If a column, subjected to a longitudinal force, be cut out of a
plank or slab of equable depth, in order that the extension and com
pression of the suifaces may be initially every where equal, its outline
must be a circular arc.
CHAP. VIII.] MODULUS OP ELASTICITY. 383
If a column be cut out of a plank of equable breadth, and the
outline limiting its depth be composed of two triangles, joined at
their bases, the tension of the surfaces produced by a longitudinal
force will be ererj where equal, when the radius of curvature at the
middle becomes eaual to half the length of the column ; and in this
case the curve will be a cycloid.
When the curvature at the middle differs from that of the cycloid,
the figure of the column becomes of more difficult investigation. It
may, however, be delineated mechanically, making both the depth of
the column and its radius of curvature proportional always to ^/a.
If the breadth of the column vary in the same proportion as the
depth, they must both be every where as the culw root of a, the
ordinate. ( Young 9 Nat, Phil, vol. ii.)
The modulus of elasticity has not yet been ascertained in reference
to so many subjects as could be wished. Professor Leslie exhibits
several, however, as below. That of white marble is 2,150,000 feet,
or a weight of 2,620,000 pounds avoirdupois on the square inch ;
while that of Portland stone is only 1,570,000 feet, corresponding
on the square inch to the weight of 1,530,000 pounds.
White marble and Portland stone are found to have, for every
square inch of section, a cohesive power of 1811 lbs. and 857 lbs. ;
wherefore, suspended columns of these stones, of the altitude of 1542
and 945 feet, or only the ld94th and 1789th part of their respective
measure of elasticity, would be torn asunder by their own weight.
Of the principal kinds of timber employed in building and car
pentry, the annexed table will exhibit their respective Modulus of
Elasticity, and the portion of it which limits their cohesion, or which
lengthwise would tear them asunder.
Teak 6,040,000 168th
Oak 4,150,000 144th
Sycamore 3,860,000 108tli
Beech 4,180,000 107th
Ash 4,617,000 109th
Elm 5,680,000 146th
Memelfir 8,292,000 205th
Christiana deal .... 8,118,000 146th
Larch 5,096,000 121th
The following, also, exhibits Mr. Sevan s results as to the modulus
of elasticity.
Feet.
Platinum 2,390,000
Gold (pure) 1,390,000
Steel 9,300,000
Bar iron 9,000,000
Ditto 8,450,000
Yellow pine 9,150,000
Ditto 11,840,000
384 MODULUS OF ELASTICITY. [PART II.
Feet.
Finland deal 6,000,000
Mahogany 7,600.000
Rose wood 3,600,000
Oak, dry 6,100,000
Fir bottom, 25 years old 7,400,000
Petersburg deal 6,000,000
Lance wood 5,100,000
Willow 6,200,000
Oak 4,360,000
Satin wood 2,290,000
Lincolnshire bog oak 1,71 0,000
Lignum Vitae 1,860,000
Teak wood 4,780,000
Yew 2,220,000
Whalebone 1,000,000
Cane 1,400,000
Glass tube 4,440,000
Ice 6,000,000
Limestone.
„ Dinton, Buck 2,400,000
„ Kctton 1,600,000
„ Jettemoe 636,000
Ryegate 621,000
Yorkshire paving 1 ,320,000
Cork . 3,300
Slate, Leicestershire 7,800,000
The following is the weight of the modidus of elasticity of various
substances employed very generally in construction, according to Mr.
Tredgold.
Pounds.
Ash 1,640,000
Beech 1,346,000
Brass, cast 8,930,000
Elm 1,340,000
Fir, red or yellow 2,016,000
Fir, white 1,830,000
Iron, cast 18,400,000
Iron, malleable 24,920,000
Larch 10,740,000
Lead, cast 720,000
Mahogany 1,696,000
Oak, good English 1,700,000
Pine, yellow American 1,600,000
Steel 29,000,000
Tin, cast 4,608,000
Steel, cast 13,680,000
CHAP. Vni.] TaAN8V£B8B 8TBBN0TH OP CAST IBON. 385
Sbct. IV. On the Strength of Materials svhjected to a tramverse
strain.
From the obseirations which we have made in the first section, it
appears that we are not possessed of all the data which are required
for determining generally the strength of a castiron heam of any
given form of section ; and consequently, that although rules have
been given for that purpose, their results are in many cases, (as might
be expected,) at utter variance with the actual strengths as obtained
by experiment. Such, then, being the state of our knowledge upon
the transverse strength of materials, we have abstained from giving
any rules deduced alone from theory^ being well assured that to have
an erroneims rule is much worse than to be without ani/ rule at all.
Nevertheless, the strengths of a few forms having been determined
by actual experiment, we are enabled to calculate with certainty, by
means of formula (HI.), page 374, the strength of a beam of any
dimensions, whose form of section is similar to any of those forms,
using the word similar here in its strict geometrical sense, as explained
in definition 12, page 103.
1. For the rectangular form of section :— From the mean of 265
experiments of Messrs. Hodgkinson and Fairbairn, it appears that a
weight of 454*4 lbs. applied at the center of a bar of cast iron 1 inch
square and with a bearing of 4*5 feet produced fracture ; therefore,
for a bar of any other dimensions, putting W for the breaking weight
applied in the center, L the length of the beam in feet, h and d its
breadth and depth in inches, we have
2045 6rf' _ . ,^
= W, m lbs.
L
18'25^(/^
L
•912 b d'
= W, in cwts.
=s W, in tons.
(VIII.)
2. For Mr. Hodgkinson's form of section (fig. 235) in which the
*i%a (a) of the lower flange is made six times that of the upper, wc
We
4862 ad _ . ,,
= W, m lbs.
L
4833 acf
L
2166a</
= W, in cwte.J (IX.)
= W, in tons. I
S8S TRAirSVEBSB 8TBSN0TH OF CAST IRON. [PART II
3. When the form of section does not materially differ from that
shown in figure 235, the following formula is found hy Mr. Hodg
kinson to agree moderately well with the results of experiments. If
h^ equal the entire hreadth of the bottom flange, b^ the thickness of
the vertical part, d^ equal the whole depth of the girder, d^ the depth
without the lower flange, all in inches, and L equal the length in
feet, we have
^^^^ {^ ^i'  (*i  ^i) ^1 = W, in lbs. "*
<L
40
{*i ^i'  (^  ^2) ^2*} = W, in cwts. > ... (X.)
I
4. We have also been favoured with the following arbitrary for
mula by Mr. Dines, which he has found to be tolerably correct in
all cases where the length of the girder did not exceed 25 feet, its>
depth in the center was not greater than 20 inches, nor the breadth
of its bottom flange less thim onethird, nor more than half th&
depth, and the thickness of the metal not less than ji^th of th^s
depth. Then the letters expressing the same quantities as Hpfnw*
we have
170^ "V
j {«, d,"  (i.  b,) dn = W, in lbs.
80
5l7 {*i •'i*  (*,  *,) ''/} = W, in cwU. ... (XI.)
~ {4, rf.»  (J.  h) «/«*} = W, in toBs.
LONOrrUDINAL FORM OF BEAM OF UNIFORM STRENGTH.
1. I/the depth (^tke beam is uniform: —
When the whole load is collected in one pohit, the sides of th^'
beam should be straight lines, the breadth at the ends being half thi^^
where the load is applied, as in fle. 286.
When the load is uniformly dntributed, the sides shookl be pos^^
tions of a circle, the radius of which should equal the square of tli^v
length of the beam, divided by twice its breadth, as in fig. 237.
2. When the breadth of the beam i» uniform: —
When the load is collected in one point, the extended side of ttr^
beam should be straight, the depth at the point where the load ^
applied twice that at we ends, and the linea connecting them straight,
as in fig. 238.
CHAP. VIII.] TBAN8V£R8B 8TBEN0TH OP MATERIALS.
387
When the load is uniformly distributed, the extended side should
be straight, and the compressed side a portion of* circle whose radius
equals the square of half the length of the beam divided by its depth,
as in fig. 239.
3. When the transverse section rf tie beam is a similar figure
tknmghout its whole length: —
When the load is collected ib one point, the depth at this point
should be to the depth at hm extremities as 3 : 2 ; the sides of the
beam being all straignt lines, as in fig. 240.
When the load it vniformly distributed, the depth in the center
should be to tba depth at the end as 3 : 1 ; the sides of the beam
being all stnHJgbt lines, as in fig. 241 .
mANSYKBSE STBENOTH OP OTHER MATERIALS THAN CAST IRON.
The only form of beam which is employed of any other material
than castiron is the rectangular form, the strength of which may be
immediately obtained from the following formula, in which h is the
breadth, d the depth, both in inches, L the length in feet, and W the
breaking weight applied in the center ; then
?i^ = W,mlb8.
li
L
o^bd'
= W, in cwts.
= W, in tons.
(XII.)
The following table exhibits the values of the constant coefiicients
^^ a,, and a.^ according to Professor Barlow, for several different
i^nds of timber.
Name of Material.
Value
of a.
Value
of a^
Value
of a^
Authority.
brought iron
Brass, cast ...
I«ead, cast
•nn,caat
Zinc, cast
Teak
Poon
Ash
Canadian oak.
English oak...
I^tch pine.....
2290
890
196
872
746
821
740
676
589
567
544
2046
796
176
332
666
783
661
603
626
498
486
•022
•397
•087
•166
•338
•366
•380
•301
•263
•249
•243
Tredgold.
Barlow.
c c 2
388
ELASTICITY OF BODIES.
[PABT II.
Name of Material
Beeclj
Dantzic oak
Adriatic oak
Red pine
Mar Forest fir....
New Eugland fir.
Riga fir
Elm
Larcb
Value
of a,.
519
486
461
447
408
367
359
338
330
Value
Valae
of a^
of aj.
Authority.
463
•232
Barlow.
433
•216
jy
412
•206
>1
400
•200
>^
364
•182
99
328
•164
99
321
•160
99
302
•151
99
294
•147
99
THE STRENGTH OF BEAMS ACCORDIXO TO THE MANNER IN WHICH
THE LOAD IS DISTRIBUTED.
In the same beam, the weight which will be required to break it
depends very much upon the point at which it is applied ; and upon,
the ^ay in which it is distributed ; as also upon the manner in whid^
the beam is supported.
1. When the beam is supported at both ends : —
If the weight which must be applied at the center to ^
produce fracture, equals ) •
Then that which must be applied at any other point, \
(where /j and l^ equal the distances of that point from r L' —
each support, and L the distance between the sup I ^ I I
ports,) ii^ill be equal to ;
And that which would be required if distributed ) o w
uniformly along the beam, would be equal to J
2. When the beam is supported only at one end : —
If the weight is applied at the other, it must be ) i t^ .
equal to J * '
And if distributed uniformly along the beam, it must ) i »r
be equal to J *
Sbct. V. Elasticity of Bodies subfected to a transverse strain,
The deflection of rectangular beams when supported at each exL 
mity and loaded in the middle, is found by Uie following fonnu ^
in which i equals the deflection in inches, Wj the weight in Iba. pr" "^^
ducing it, and m the weight of the modulus of elasticity in lbs. ^^^
the given material, which is given at pages 38? and 384 ; the oUt^^^
letters represent the same quantities as before, tlien we have
'=i^^ («"■)
CHAP. VIII.] ELASTICITY OP BODIES. 389
If tbe weight be uniformly distributed, instead of being collected
in tbe center, it will only produce th8 of tbe deflection given by tbe
above formula.
For castiron girders whose dimensions are limited, as described
in § 4, page 386, Mr. Dines finds that the following formulee may
be made use of to determine the deflection of the beam when
loaded in the center with ths of the breaking weight ; in which d is
the depth of the beam in the center in inches, and L the length in
feet.
When the top and bottom flanges are equal, and the girder
parallel, or of equal depth throughout —
* = i^ .(^^^^
When the flanges are not equal, and the girder is not parallel —
* = ^. (^^J
When the beam has no top flange, and the depth varies, then
^ = 8^^ (^VI)
Thefte formulee are the result of upwards of 2000 experiments
upon beams of a moderately large size.
In the first section we stated that the elasticity of a castiron beam
becomes injured with a strain very much smaller than the breaking
weight, so that, when the strain is removed, the beam does not
recover its former shape, but remains permanently bent. Mr. Hodg
kinson was, we believe, the first writer who noticed that such was
the case, which he has proved by his experiments, and which result
has been fully confirmed by those of Mr. Cubitt.
The greater part of Mr. Hodgkinson's experiments were made
apon beams having only a bearing of 4 feet 6 inches ; and he found
that the amount of the permanent set varied as the square of the
weight applied. This rule, however, although correct for beams
about 5 feet in length, does not apply when the length becomes
mnch greater, for Mr. Cubitt found by his experiments, that when
the length became equal to about 20 feet, the set was only as the
weight, and that with larger beams the set was still less. As these
exponents are important from the size of the girders, we have
tabulated a few of them, as on the following page.
390
MR. GUBITT8 SXPBBIMBNTS.
[part II.
1 n
n
1 i ij
1 ip
1
BBM^ftKa.
L
2
i
8
10
14
18
"000
172
206
'30fi
■4B0
iia4
7S2
sa4
■100 ^ 0070
■100 Zim '^^
■006 ,S.m ' ^^^
102 ^]l mm
The form and dimeii«Joni of
theae girdezi are ahon*!! in %«
242. The result here gtTen are
tbe mean of two gif4et%, one of
which weighed 41 cf»t, qn. 20
Ibs.^ Attd the other 41 cwt. 2 qrt.
6 Ibft. The (otU length of the:
girders ww 30 fl. 4 in., the kngth
of bearing 28 ft.
^
3
4
10
12
14
IG
IB
Ofll
121
■221
305
'300
ao3
005
'70»
300
oil i^lJS
'^'^^ j*0800
'0050 1 Vm% from tUe wmje jmntsni Uj
0050 No.t The rwulta are the moml
0045 uf two girdent, one of which
<W70 weighed 42 cwu qn. 14 IbfcJ
0105 the other 41 cwt. 3 qrt. 16 Ibt^
0105 The length of beftring wma 37 £tM
11165 ?«**^ 1^
3.
2
4
6
8
10
12
•117
213
'406
096
133
•120
124
•lie
020
025
•032
04$
•057
068
^2 The result* arc the oieAn of two
jjri girders, One of which ireighed 26
^\l Wt. Iqr. 26 lb*,, and the olher
;?{* ,25 cni. 2 qm 22 lb«. The tengtb
"* * of the girders wm 24 ft, 6 In., the
ilen^ o( beanng 33 ft 10 in.
^^
4.
%
4
6
B
10
11
125
'374
H03
012
071
'140
■log
110
110
m9
0150
0245
0350
mm
'0620
mm
1 Cam from the wiQe pfttUifn ■«
0095 'Nil. 3. The faults Are the ni«ftn
■0105 of two girdeiTft, one of vhich
■0130 weighed 2&cm, 2 qr«, Iht, uid
0140 the oiher 24 cwt. 3 qn, I7 Ibsk.
0070 The lengtb of the be»riBg vm
23 ft.
ft.
4
a
10
13
'24
'38
'50
'64
74
12
14
14
•10
•035
050
065
•HBO
1 The fonn and d]aie»i»k»ni of tb^^^
01 ft prd*r ajneihown in hg. 244, Th* ^^
rtiR 1 weight of the giMer wm 10 cwt*^ ^^
;?{* (0 qr». \B lbs., its total length ^r^
^*^ Ifeet, and the distftnce betvup. ' ■
[the bcftrLngm 20 ft. 7 iru
6.
1
2
3
4
4i
085
4:4
380 1
402
475 1
0«9
100
123
073
018
035
•000
•103
013 '
m&
■000 ;
■017
043
1123
E«ctiuiguliir bar of out ln»=3
51 inches in depth* 1^04 inditf
in breadtht and 5 ft, 6 in, lengC
of bearing. Broke at & tons.
f
7.
1
3
4
44
083
16!
360
'300
'440
•078
OtfO
120
•060
015
1Ki2
•030
RecL^ngulELr bur of emm iraiiw
Bi incliei to depth, I 06 incMr
in br^dth, and 5 feel 6 ipcha
lengUi of b«mng. Broke v%^
41 tonA^
CHAP. VIII.]
MR. CUBITTS EXPERIMENTS.
391
i
d
1^
^1
11
Q
h
I*'
1
a
mmrABKii
' 1
2
3
4
ft
6
11*7
100
146
105
•250
350
063
045
060
066
100
005
009
■020
■lao
■004
■Oil
■160
RecUngulAr tmr of wucmght
Iron, 4 '97 mdies in depth, 1 inch
in breath, and 5 ft 6 in. length
of be&ring.
1^
1 1
1
S
3
4
ft
6
060
110
166
210
300
630
*^* No tat,
0^ i 000
OfrS 1 012
090 5541
230 ! 260
■006
il3a
■2O0
Kecungulfir b^r of wrotight
iron, 5 '03 incbeii in depth* 1 IqcIi
in breadth. And 6 ft, 6 in* lin^
of bearings
11K
S
3
4
G
100
146
•206
402
but
046
060
197
130
Trifling
■01 a
■154
'280
■013
130
■126
Rectangul^ bar of wrought
iron, 6 inc&Bi m depth, '97 incheii
In breadth, and 6 ft* 6 in. length
of bearing.
11.
3
3
4 .
ft
loe
160
318
310
380
■062
■oeo
093
*070
Trkfling
■0O8
■044
•106
■012
<>24
il6l
Hecuingular hsr of wrought,
iron, 4'97 inches In depth, 102
inchefl iti bresfitbi aod 6 It. 6 In,
length of bearingp
IS,
1
3
3
' 4
•060
106^
160
200
238
26a
■066
i)46
■066
032
■030
Set.
004
■008
■016
■020
■028
0O4
■007
006
008
Rectangular bar of wrought
iron, 4 97 mckim in depth, 1^02
inches in breadth, and 6 fu 6 in.
length of bearing.
13.
1
3
3
4
41
060
103
148
•200
230
063
•046
■062
•020
Set
■004
BectJingular bar of wrought
iron, 4'»7 *nche« in depth, 102
inchei in breadth, and 6 fU 6 in,
length of bearing.
14.
I
li
3
n
3
Si
4
•162
■302
'348
■300
■374
ma
■060
046
062
074
■002
■003
008
■010
■066
•306
•001
005
•002
046
■161
Rectangular bar of wrought
iron, 4 Inches in deplh^ 1^01
inches in brewJth^ and 6 f^. 6 in,
length of bearing.
IS.
I
i'
4
41
■100
150
loa
296
'3^
■460
G&8
4)40
046
•oeo
■OGO
■006
'208
^103
•010
■016
■023
046
102
■266
mi
006
■007
■023
•066
163
Rectangular bar of wrought
iron, 4 incbea in depth, I 01
Ini^ei in breAdth, and 6 ft 6 in.
length of bearing.
392
STRBNOTH OP MATERIALS.
[part II^
Sbct. VI. Strength of MaieriaU to resist Torsion.
The power of a bar or rod of any material to resist torsion, maj^
be measured by the angle through which the end of a lever attacbec^
to the same will be moved by a given weight. If D equal the dis^
tance from the fixed end of the bar or prism to the point of applica^i
tion of the lever used to twist it, / the length of the lever in inches
w the weight in lbs. applied to the end of the lever, r the radius c^
the pnsm if cylindrical, and 5, d^ its breadth and depth if rectangulau4
all in inches, § the angle of torsion at the point of application, F a cona
stant for each material, representing the specific resistance to flexur .
by torsion, and W a constant weight expressing the resistance to to ^
sion, with regard to a unit of surface at the time of fracture ; th^
the following table will exhibit the relations subsisting between thc^
quantities.
Form of section of
the prism.
Ketistanoe to angular flex
ure by a force of torsion.
Resistance to fracture I
a force of torsion.
Cylindrical .
Square ....
Rectan":ular
F = »/
2L
W
F=«/i(f±^)i^
wl
2_
e
The mean value of W for cast iron, as obtained from the experi
ments of Messrs. George Rcnnic, Bramah, and Dunlop, is 32,503 lbs.,
and from Mr. Bevan's experiments, the mean value of F for cast iron
is 5,709,600 lbs., and for wrought iron and steel J 0,674,540 Iba^
and, these values being substituted in the above table, it becomes
PonnorMclion
of the prism.
Cylindrical
Square
Rectangular
Retlftance to angular flexure by a force of tornoo.
. Cast iroD.
u»/  8968620— .
tr/96I600
d*0
v/» 1903200
b*dU
(6« + rf«)L
Wrought iron or steeL
ir/= 16767770—.
d*i
^/ 1779090—.
Rcristance to ftactwc'
aforceoftontoa.
Cattfaon.
w/ 51066 r«.
w/»76Clif*.
IV/108S4
APPENDIX.
TABLE OL
[Ho, Um h. 175802
t
s
4
5
«
7
S
1 «
N.
ISl
ISQl
8876
llHO
B*2U
1734
8038
2188
6468
2598
8894
3029
7321
3461
7748
3S01
8174
100
1
2
3
4
6
8
7
8
9
no
1
2
8
4
8
7
8
9
120
1
2
3
4
5
8
7i
8
9
ISO
1
2
3
4
5
6
7
3
140
1
2
3
4
5
6
7
8
m
0300
4521
8700
0724
4940
9116
1147
5360
0532
1670
5779
0047
1993
6197
2415
8616
\m
0361
4488
8571
0775
4896
S078
m
242S
6533
2841
6042 '
3252
7350
3664
7757
4075
8164
1^5
*S7
060O
462S
8620
3
1004
5029
0017
4
1408
5430
9414
5
18)2
5830
9811
6
2218
1230
2619
6629
3021
7028
£23
8 1
0207
7
4148
8053
0602
8
4540
8442
0098
4038
8830
M
M
2578
6495
03S0
4^230
3048
2989
6885
3362
7275
8755
7664
{46
0768
4613
8428
1153
4006
8805
1538
5378
9185
1024
5760
0563
2309
6142
0042
2694
6524 '
M»ti
0320
4088
7816
152
106
1829
5580
9298
2206
5953
9658
2582
8326
2058
8609
3333
7071
3709
7448
m^
0038
3718
7363
5
0407
4085
7731
6
0776
4451
8004
7 1
1145
4818
8457
8
1514
5182
8819
m
rre
8
29B5
6640
3
3352
7004
4
m
0206
3881
7426
0826
4210
7781
0087
4576
8138
1347
4034
8400
1707
5201
8848
2067
5647
0198
2428
6004
0552 '
Ul
22
0983
4471
7051
1315
4820
8208
1667
5160
8644
2018
5518
8990
2370
6866
9335
2721
6215
9681
3071
6562
HI4
0028
3462
6871
1403
4828
8227
1747
5180
8585
2001
5510
8903
2434
5851
0241
2777
6191
9570
B119
6531
9018
m»
0253
3609
6040
:63
8
in
1509
8
4644
8265
1934
4
5278
8596
2270
5
5811
8926
2605
6
5943
9256
2940
7
6276
9586
3275
B
6608
9915
m
0245
3525
6781
mi
m
1560
4830
8076
12&8
4406
7671
1888
5156
8300
2216
5481
8722
2544
5S06
9045
2871
8131
9368
8198
6456
9600
£»3
0012
3219
6403
9564
,77
77
1819
4814
7087
1030
5133
8303
2260
5451
8618
2580
5760
8034
2000
8086
9249
m
m
48
0822
3051
3
7058
1136
4263
4
736T
1450
4574
6
T676
1763
4835
6
7085
2076
5196
7
8204
2380
5507
8
8603
2702
6818
9
8911
m
»43
0142
3205
6246
9266
0440
3510
8549
9567
0756
3815
8852
9868
1063
4120
7154
1370
4424
7457
1676
4728
7750
1082
6032
8061
»66
0168
3161
6134
9086
0489
3480
6430
9S80
0769
3753
6728
0874
1068
4055
7022
0988
m
'OS
2266
5244
8203
2564
6541
8497
2863
58S8
8792
148
eft
IHl
4080
H34
4351
1726
i 4641
2019
14932
2311
2803
6512
2696 ;
5803
*B
K(>. l&OO L. 170091]
TABLB III.
[N«. 1999 L. 30031
1
m
1
s
3
*
9
160
1
, 2
a
4
1 5
1
2
3
4
S
6
r 1
a
e
s
176091
6381
6670
0652
6969
9839
724S
7536
7825
8113
8401
8689
15
8977 i*ao*
0126
2985
5825
8647 '
0413
3270
0103
8928
0609
35^5
6391 1
0209
0986
38^9
6674
9490
1272
4123
6956
9771
1568
4407
7230
1;
2
S
4
5
«
7
SOD
1
2
1
S
!
i
1
»
7*
1
1
a'
1S1S44
4 SOI
7521
2129
4975
7803
2415
6259
80S4
2700 1
5542
8366
msi
2346
5623
£382
190332
3125
8667
0612
3403
6170
0892
6453
1171
3959
6720
H81
1461
4237
7005
9755
1730
4514
7281
2010
4792
7556
2289 : 2667
6009 1 5346
7832 8107
8932
9200
0029
2701
6
6475
8173
0303
3033
6
5746
3141
0577 ; 0S5O
11^4
3346 f
.^'
924TI
a*l397
4120
6326
16T0
1
4391
7096
1»43
2
466S
7366
2216
%
4934
7634
2488
4
6204
7004
3305
7
6016
8710
^77
8
6236
3979
9515] Ui&a
0061
1*319
2980
5638
8273
0586
3252
5902
8530
0353
3518
6160
3793
1121
3783
6430
9U0O
1368
4049
6694
0323
1054
4314
6957
9535
IU21 !
45711 ''
7221
9845
212lB8i 2464
4844' SI 09
7484 7747
2720
5373
8010
! 6
7
S
,220108 0370 0631
1 2716 2970 \ 3236
£J09 5503 1 5826
0892
3490
0U84
8667
1153
3755
6342
3913
1414
4U15
6000
9170
1075
4274
6858
9420
193«j
4533
7115
9682
2190
4792
7372
0938
245ff
5051
76S0
9
m
1
3
7887 1 8144 1 8100
am .
1 '
2742 1
5276
7793 ,
1
230440 0704
2996; 3250
£528 5781
2
OOtiO
3604
6U33
8548
3
1215
3767
6285
8799
4
1470
4011
6537
9049
5
1724
4264
6789
9299
1979
4517
7U41
1 9550
7
2234
4770
7292
0800
i
243S
5023
7544
8046' 8297
0050
2541
5lM9
7482
9032
om
4
e
7
240549' 0799
3038 3280
6513 6759
7^73 8219
1U48
3534
60U6
8464
1297
3782
6252
8709
1540
4030
6499
8954
1795
4277
6745
0198
2044
4525
6991
9443
2293
4772
7237
9087
27*6 1
5M J
8
ISO
1
25U420
2853
<
5273
7679
U604
3096
I
&614
7918
0^08
S338
2
5756
8158
1161
3580
3
6090
3398
1395
3322
4
6237
8637
1638
4064
6
6477
8877
1081
4306
6
0713
9116
2l:£5 i 2:163
4548 1 4790
7 ! «
6963 7193
9355 9594
20l«)
5031
9
7431?
a
m
I
. t
i
260071
2461
4818
7172
0613
0310 0548
2688 1 2925
6U64 52^0
7400 7641
9740 9080
0787
3162
5525
7876
1025
3399
5761
3110
1263
3630
5990
3344
1601
3873
6232
8678
1739 1970
4109 4340
6467 6702
3812 1 9040
1214
0213
2538
4350
7151
3
9439
iHl46
2770
50B1
7330
; 4
9067
Otfr9
3001
5311
7609
5
9895
0912
3233
5542
7838
6
1144
3464
5772
8067
7
1377
3606
0002 i
S296
8
im
V
8
9
271842
4158
6163
2074
4389
0692
1
2ay(J
46:i0
0921
2
39iT
190
8754
8932 ; 9211
0123
2306
4056
6905
9ua
0351
2022
4832
71 SO
9366
0578 1
2849
6107
T354
9539
"«Wi
m
1
t
3
2810331 1201 ' 1488
3301 3527 ^753
5657 5782 6007
7S02I 8023 8249
1716
3979
6232
8473
1942
4205
6456
8696
2169
4431
6631
3020
m
m
m
1
i
i
i
1
i
1
5
7
S
9
290OJJ5
2266,
4400
6666
8863,
0267
2478
4687
68^4 i
0071
0480
mn
4907
7104
0289
0702
2920
5127 1
7323
9507
0925
S141
6347
7542
9725
1147
3363
5567
7761
0043
1309
3584
5787
7979
1591
3304
6007
3198
1313
4026
0226
8416
i
1
t
0161
0373
0696
1 oau
1
V%i0io 1^0010301
TABLB IIL
[N*. 2499 J,. 897766
1 1
2
3
4
6
1 T
&
9
1
S
8
4
6
6
7
8
9
K.
301030
S19@
7490
M30
311754
S3&7
5^0
1247
8412
8566
7710
0843
wm
4078
6180
8S72
1464
3628
8781
7924
1681
3844
5996
8137
1398
4069
6211
8361
2114'
4275
6425
8564
2331
4491
6639
8778
2647
4706
6854
8601
2764
4921
7068
9204
2980
5136
7282
9417
2oa:
li
2
3
4
5
6
7
8
6
0066
2177
4239
6390
8481
0263
2380 ,
4490
6699
8689
0481
2600
4710
6809
8808
0693
2812
4020
7018
9106
0906
3023
5130
7227
6314
1118
3234
5340
7436
9522
1330
3446
5561
7646
6730
1542
3656
5760
7854
9938
320146
0354
05^2 j
0769
0977
1184
1391
1593
1805
2012
1
2
3
1
5
! d,
r
2210
4282
6336
1
2426
4488
6541
2
2633
4694
6746
8787
3
2839
4869
6950
8991
4
3046
5105
7155
9164
5
3252
5310
7359
9308
6
3458
5516
7563
9601
7
3665
5721
7767
0806
8
3871
5926
7972
9
4077
8181
8176
210
1
2
l^3S0 nsh^
0008
2034
4051
6059
8058
0211
2236
4263
6260
8267
3
330414 oeir ;
2438 S64(»
4154 16^
6460 6660
84^6 86M
0819
2842
4856
6860
S855
1022
3044
6057
roflo
9064
1226
3246
6257
7260
9253
1427
3447
6458
7459
9451
1630
3649
6658
7659
6650
1832
8860
5869
7866
9849
41
6
6,
T
U047
2028
0246
fl
S40444 064^
0841
1030
1237
1436
1632
1830
2226 9
1
s
1
!
2423
43&2
e353
S30j^
1
2820
4689
6549
8600
2
2817
4786
6744
8094
3
3014
4081
6039
8SS9
1
3212
5178
7135
9083
5
3409
6374
7330
9278
6
3606
5570
7525
0472
7
3802
5766
7720
9666
8
3969
6662
7615
9860
9
4196 2^0
6167 1 1
8110 1 a
0064 3
350^48
218S
4108
eo2«
7935
»835
0442
S375
4301
8217
81 35
0636
2668
4493
6403
8316
0829
2761
4686
6699
8606
1023
2654
4876
6790
8696
1216
3147
6068
6981
8886
1410
3339
6260
7172
6076
16i>3
3632
5462
7363
9266
1708
3724
5643
7554
9456
1689
3910
5834
7744
9646
4;
6
6
7
8
9
oose
021^
040^
0593
0783
0972
1161
1350
1539
L
1
3
3
1
301728 1»17
M12: 8800
W88' fi675
736« 7543
9218 9401
210#
8938
6862
77^
9687
3
2294
4176
8049
7015
©772
4
2482
4363
6236
8101
9658
6
2871
4551
6423
8287
0143
1091
3831
5664
7488
9808
6
2859
4739
6610
8473
0328
2176
4016
5846
7670
9487
7
3048
4626
6796
8659
0513
2360
419B
6026
7862
9668
8
3238
5113
6083
8845
0698
2544
438?
6212
mu
9849
6301
7169
6030
08S3
330
I
3
4
•
37l0fiS 1263
2912' 3096
4748 4932
8677, 6769
8S98' 8630
1437
3280
5115
6942
8761
1622
3484
5298
7124
8943
JbOe
3647
5481
7308
9124
2728
4566
6364
8216
5
8
9
0030
9
1837
3636
5428
7212
8989
9
240
1
S
8
4
5
6
r
8
9
»1T
8815
M06
r»90
i
038^
2197
8995
8785
75$8
0673
2877
4174
5964
7746
9620
3
0754
2557
4363
6142
7924
9698
4
0634
2737
4533
6821
8101
0876
5
1115
2917
4712
6499
8279
6
1296
3097
4891
6677
8456
7
147.6
3277
5070
6356
8634
8
1666
3456
6249
7034
8811
\ '
9166 9348
0051
1817
3575
6326
7071
0223
1993
3761
5501
7245
0406
2166
3936
0582
2345
4101
6850
7592
0769
2521
4277
6026
7766
y
39093^ 1112
i8»7 2878
4452 4627
Bim 08T4
1238
3048
4802
6^8
1464
8224
4977
! 67i2
1841
3400
6152
6396
*Ba
No. 2600 L. 397940]
TABLB III.
[No. 2999 L. 476976
8461 I 8634
2 401401
8 3121
4 4834
5 6540
6 8240
7 9933
6710 I 6881
8410 8579
0365
2089
3807
5617
7051 I 7221
8749 ' 8918
8808 , 8981
9154
0538 0711
2261 2433
3978 I 4149
5688 5858
7391 I 7561
9087 ! 9257
8 411620
9, 3300
260 4973
1 6641
2 8301
8 9956
4 421604
5 3246
6 4882
7 6511
8 8135
9 9752
0102 0271 0440 ' 0609 ! 0777 0946
1788 j 1956 I 2124 2293 2461 ' 2629
3467 3035 , 3803 3970 4137 ! 4305
1
5140
6807 i
8467 '
I
6
2 ' 3 4
5307 6474  5641 6808 , 5974
6973 7139 , 7306 I 7472 ' 7638
8633 8798 8964 , 9129 9295
0883
2605
4320
6029
7731
'N.
9328 ' 9501 250
1
1056 , 1228
2777 2949
4492 4663
6199 ' 6370
7901 8070
9426 9595 9764
1114 1283
2796 2964
1451
3132
4472 I 4639 ' 4S06 ] 9^
8
6141 ' 6308 , 6474 260
7804 i 7970 , 8135
9460 9625 ' 9791
1
0121
1768
3410
6045
6674
8297
9914
0286 0451 0616
1933
3574
5208
2097 I 2261
3737 3901
6371 5534
6836 I 6999  7161
8459 8621 8783
0075 j 0236 i 0398
270 431364'
1 2969
2 4569!
3 6163
7751
9333
4
6
6 440909
7 2480
8 4045
9, 5604
280 7168
1 8706
1
1526
3130
4729
6322
7909
9491
1685
3290
4888
6481
8067
9648
8
1846
3450
6048
6640
8226
9806
4
2007
3610
5207
6799
8384
9964
1066
1224
1381
1538
2637
2793
2950
3106
4201
4357
4613
4669
6760
6915
6071
6226
1
2
8
4
7818
7468
7623
7778
8861
9015
9170
9324
2 450249
8, 1786
41 3318
6 4845
6 6366
71 7882
8 9392
0865 ,
2400 I
3930 !
5454
6973 ,
8487
9995
9 460898
290 2898
li 8893
2 6883
8 6868
41 8847
6 9822
0657
0704
2026 2171
8487
8638
4944
6090
6897
6542
. 8000 L. 477121]^
TABLE III. [No. 3499 L. 543944
1
2
3
4
5
6
7989
7
8
1
9
N.
800
) 477121
7266
7411
7555
7700
7844
8133
8278
1 8422
L 8566
8711
8855
8999
9143
9287 1 9431
9575
9719
9863
1
2
I 480007
0151
0294
0438
0582
0725
0869
1012
1166
1299
\ 1448
1586
1729
1872
2016
2159
2302
2446
2688
2731
3
I 2874
3016
3159
3302
8445
3587
3730
3872
4016
! 4157
4
> 4300
4442
4585
4727
4869
5011
6163
6296
5437
6679
5
5 5721
5863
6005
6147
6289
6430
6572 i 6714
6866
6997
6
' 7138
7280
7421
7563
7704
7845
7986
8127
8269
8410
7
\ 8551
8692
8833
8974
9114
9255
9396
9637
9677
9818
8
9
> 9958
0099
1
0239
2
0380
3
0620
4
0661
5
0801
6
0941
7
1081
8
1222
9
) 491362
1502
1642
1782
1922
2062
2201
2341
2481
2621
310
2760
2900
3040
8179
3319
3458
3697
3737
3876
4015
1
I 4155
4294
4433
4572
4711
4860
4989
5128
6267
5406
2
I 5544
5683
5822
5960
6099
6238
6376
6516
6663
6791
8
^ 6930
7068
7206
7344
7483
7621
7759
7897
8036
8173
4
i 8311
1 9687
8448
9824
8586
9962
8724
8862.
8999
9137
9275
9412
9550
5
6
7
0099
1470
0286
1607
0874
1744
0611 1 0648
0785
2164
0922
2291
501059
1196
1333
1880
2017
2427
2564
2700
2837
2973
3109
3246
3382
3618
3656
8
8791
3927
4063
4199
4335
4471
4607
4743
4878
6014
9
1
2
3
4
5
6
7
8
9
5150
5286
5421
5557
5693
5828
5964
6099
6284
6370
820
6505
6640
6776
6911
7046
7181
7316
7461
7686
7721
1
7856 7991
8126
8260
8395
8530
8664
8799
8984
9068
2
9203
9337
9471
9606
9740
9874
0009
1349
0148
1482
0277
1616
0411
1760
3
4
510545
0679
0813
0947
1081
1215
i 1883
2017
2151
2284
2418
2561
2684
2818
2961
3084
5
1 8218
3351
3484
3617
3750
3888
4016
4149
4282
4415
6
' 4548
4681
4813
4946
5079
5211
5344
6476
6609
5741
7
( 5874
6006
6139
6271
6403
6535
6668
6800
6932
7064
8
» n96
7328
7460
7592
7724
7855
7987
8119
8261
8382
9
1
2
3
4
5
6
7
8
9
> 8514
I 9828
8646
9959
8777
8909
9040
9171
9303
9484
9666
9697
330
1
2
0090
1400
0221
1530
0353
1661
0485
1792
0615
1922
0745
2053
0876
2183
1007
2814
S 521188: 1269
\ 2444
2575
2705
2835
2966
3096
3226
3366
3486
3616
3
I 8746
3876
4006
4136
4266
4396
4526
4666
4785
4915
4
> 5045
5174
5304
5434
5563
5693
5822
6961
6081
6210
5
S 6339
6469
6598
6727
6856
6985
7114
7243
7372
7601
6
r 7680
7759
7888
8016
8145
8274
8402
8531
8660
8788
7
J 8917
9045
9174
9302
9430
9569
9687
9815
9943
0072
1361
8
9
> 530200
0328
0456
0584
0712
0840
0968
1096
1223
1
2
3
4
5
6
7
8
9
) 1479
1607
1734
1862
1990
2117
2245
2372
2500
2627
340
I 2754
2882
3009
3136
3264
3391
8518
3645
3772
3899
1
I 4026
4153
4280
4407
4534
4661
4787
4914
5041
5167
2
) 5294
5421
. 5547
5674
5800
5927
6063
6179
6806
6432
3
1 6558
6685
6811
6937
7063
7189
7316
7441
7667
7693
4
S 7819
7945
8071
8197
8322
8448
8574
8699
8825
8951
5
Jl^9076
9202
9327
9452
9578
9703
9829
9954
0079
1830
0204
1454
6
7
r 540329
0456
0580
0705
0830
0955
1080
1205
Bl 1679
1704
1829
1953
2078
2203
2827
2452
2576
2701
8
r. 2820
8950
3074
8199
3323
3447
8571
3696
9820
8944
9
No. 8500 L. 544068]
TABLE III.
[Xo. 8999 L. 601951
N.
8
850,544068
1 6307
2 6548
8 7775
9003
4192 I 4816
5431 I 5555
6666 6789
7898 ' 8021
9126 9249
4440
5678
6913
8144
9371
4
5
6
7
8
4564
4688
4812
4986
5060
5802
5925
6049
6172
6296
7086
7159
• 7282
7405
7529
8267
8389
8512
8635
8758
9494
9616
9739
9861
99S4
9 S.
5188 350
6419 1
7652
&SS1
550228
1450
2668
3883
5094
860
1
2
8
4
5
6
0351 0478
1572 , 1694
2790 ' 2911
4004 4126
5215 5836
! 1
6303 6423
7507 7627
8709 8829
9907 "0026"
2
6544
7748
8948
0595
1816
3038
4247
5457
3
6664
7868
9068
0717
1938
3155
4868
5578
4
6785
7988
9188
0840
2060
3276
4489
I 5699
I
I 5
6905
8108
9308
, 0962
I 2181
I 3398
I 4610
> 5820
i 6
7026
8228
9428
1084
2808
3519
4781
5940
7
7146
8349
9543
1206
2425
3640
4852
6061
01i«
1828
2547
8762
4978
6182
561101 1221
2298 2412
3481 3600
4666 4784
8 5848 5966
9 7026 7144
870
! 1
I 1
8202 8319
9874 9491
0146
1340
2531
3718
4903
6084
7262
2
8486
9608
0265
1459
2650
8837
5021
6202
7379
3
8554
I 9725
0385
1578
2769
8955
5139
6320
7497
4
8671
9842
2 570543 0660
S80
1709 1825
2872 2988
4031 4147
5188 5308
6341 6457
7492 7607
8639 8754
9784
1
9898
580925
2063
3199
4331
5461
6587
7711
8882
9950
0776
1942
3104
4263
5419
6572
7722
8868
2
0893
2058
3220
4379
5534
6687
7886
8983
1039
2177 I
3312
4444 ,
5574
6700
7823
8944
0012 I
1153 1
2291 I
0061
I
591065
2177
8286
4393
5496
6597
7696
8791
9883
890
1
2
8
4
5
6
7
8
. 9 e mfsj^^
1
1176
2288
3897
4508
5606
6707
7806
8900
9992
3426
4557
5686
6812
7985
9056
0173
2
1287
2399
3508
4614
5717
6817
7914
9009
0126
1267
2404
3539
4670
5799
6925
8047
9167
1010
2174
8336
4494
5650
6802
7951
9097
0504
1698
2887
4074
5257
6437
7614
5
8788
9959
0624 , 0743
1817 I 1986
' 3006
i 4192
I 5376
6555
' 7732
1126
2291
. 3452
; 4610
! 5765
6917
8066
9212
0101
0284
3
1399
2510
8618
4724
5827
6927
8024
9119
0210
0241 I 0855
1381 ! 1495
2518 2631
3652 3765
4783 4896
5912 i 6024
7037 ; 7149
8160 I 8272
9279 1 9391
3125
4811
5494
6678
7849
6 7
8905 9023
8 9
7267 ' 7337 $60.
8469 ! 8589 I
9667 ■ 9787 2
S
4
5
6
0863 0982
2055 ; 2174
3244 , 3362
4429 4548
5612 ; 5780
6791 6909
7967 8084
8 9
9140 9257 i70i
0076
1248
2407
3568
4726
5880
7032
8181
9326
, 0193
I 1359
i 2523
] 3684
4841
5996
: 7147
'■ 8295
. 9441
03U9 i 0426 1.
i 1476 15H2 2
I 2639 I 2755 3
3800
495*7
6111
7262
8410
9555
8
0469
1608
2745
3879
5009
6137
7262
8384
9508
0583
1722
2858
3992
5122
6250
7374
8496
9615
0697 ,
1836 I
2972 '
4105 !
5235
6362 I
7486 ;
8608 ;
9726 ;
3919 i 4
6072 5
6226 «;
7877 ";
8625; f
9^ ^^
i
9 J I
0811 .8W:
1960 li
4218 5
5848 4
6475 5;
7599 5
8720 ';
9888' 9
0396 0607
4
1510
2621
8729
4834
5937
7037
8134
9228^
0819
1408
6
1621
2732
3840
4946
6047
7146
8248
9337
0428
16U
0619
6
1732
2848
8950
5056
6157
7256
8868
9446
0687
0730
7
1848
2954
4061
6165
6267
7366
8462
9566
0646
1734
0842
8
1955
3064
4171
5276
6877
7476
8672
iMAK
VvOv
0958 I 9
9
2066 m
8176 1
42S2 3:
6487! 4
7586= 5
mi , «<
9774!
0766
l«4k
0604
1961
4 "
iNoi 4000 L.
TABLE IIL
[No. 4499 L. 653116
N.
400 602060 2169
1. 8144 j 8258
4226 4834
5805 5413
68811 6489
7455! 7562
8526 8688
95941 9701
8 610660 0767
9 1728 1829
410
1
2
8
4
5
6
2784
8842
4897
5950
70001
8048'
9098
1
2890
8947
5008
6055
7105
8158
9198
2277
8861
4442
5521
6596
7669
8740
9808
0878
1986
2
2996
4058
5108
6160
7210
8257
9802
7 620186 0240
8 1176 1280
9 2214 2818
420
1
2
8
4
5
6
7
8
480
ii
4
5.
6'
1
8249 8358
4282 4885
5812! 5415
6840 6448
7366; 7468
8889; 8491
9410 9512
0844
1884
2421
2
8456
4488
5518
6546
7571
8598
9618
680428, 0580
1444 1545
2457! 2559
1
8468 8569
4477; 4578
5484! 5584
6488 6588
7490 7590
8489' 8589
9486 9586
0681
1647
2660
2
8670
4679
5685
6688
7690
8689
9686
7i640481, 0581
8: 1474! 1573
9I 2465 2568
440=
li
2!
8
4
5
6
76508^
I 1
8453 3551
4439 4537
5422, 5521
6404' 6502
7883, 7481
8360 8458
98851 9482
1S78
ffi46
0405
1375
0680
1672
2662
2
8650
4686
5619
6600
7579
8555
9530
0502
1472
2440
2886
8469
4550
5628
6704
7777
8847
9914
2494
8677
4658
5786
6811
7884
8954
I
0979
2042
8
3102
4159
5218
6265
7815
8362
9406
0021
1086
2148
4
8207
4264
5319
6870
7420
8466
9511
0448
1488
2525
0552
1592
2628
3 4
3559 3663
4591 I 4695
5621 j 5724
6648
7673
8695
9715
6761
7775
8797
9817
0783 0886
1748 1849
2761 2862
8
8771
4779
5785
6789
7790
8789
9785
4
3872
4880
5886
6889
7890
8888
9885
0779
1771
2761
0879
1871
2860
3 4
8749 8847
4784 I 4882
5717 ' 5816
6698 6796
7676 . 7774
8658 ; 8750
9627 I 9724
2608
8686
4766
5844
6919
7991
9061
2711
8794
4874
5951
7026
8098
9167
0128
1192
2254
5
8813
4870
5424
6476
7525
8571
9615
0284
1298
2860
6
8419
4475
6629
6681
7629
8676
9719
0656
1695
2782
5
8766
4798
5827
6853
7878
8900
9919
0760
1799
2835
6
3869
4901
6929
6966
7980
9002
0936
1951
2968
5
8973
4981
5986
6989
7990
8988
9984 IQOST
0021
1038
2062
3064
6
4074
5081
6087
7089
8090
9088
0978 I 1077
1970 2069
2969 3058
0599
1569
2536
0696
1666
2633
5
3946
4931
6913
6894
7872
8848
9821
0793
1762
2730
6
4044
6029
6011
6992
7969
8945
9919
0890
1859
2826
2819
3902
4982
6069
7133
8205
9274
2928 , 3036
4010 , 4118
5089 ! 6197
6166 6274
7241 7848
8312 8419
9881 9488
0341
1406
2466
7
8625
4681
6634
6686
7734
8780
9824
0447
1611
2672
8
8630
46S6
6740
6790
7839
8884
9928
0664
1617
2678
9
3736
4792
6846
6896
7943
8989
0864
0968
1903
2007
2939
3042
7
8
3973
4076
6004
6107
6032
6185
7068
7161
8082
8185
9104
9206
0032
1072
2110
3146
9
4179
6210
6238
7263
8287
9308
0123
1139
2153
3166
7
4175
5182
6187
7189
8190
9188
0183
1177
2168
8166
4143
6127
6110
7089
8067
9048
0224
1241
2255
3266
8
4276
6283
6287
7290
8290
9287
0326
1842
2366
3367
9
4376
5383
6388
7390
8389
9387
0283
1276
2267
3265
I
0016
0987
1956
2923
8
4242
5226
6208
7187
8165
9140
0382
1376
2306
3354
9
4340
6324
6306
7286
8262
9237
N.
400
1
2
3
4
6
6
7
8
9
410
1
2
3
4
5
6
7
8
9
420
1
2
3
4
5
6
7
8
9
0118
1084
2053
8019
0210
1181
2150
8116
430
1
2
3
4
5
6
7
8
9
440
1
2
3
4
5
6
7
8
9
No. 4500 L. 053218]
TABLE III.
[No. 4999 L 69888
N.
1
1
2
3
4
5
6
7
8 , 9 jir
450 653213
8309
3405
3502
3598
8695
3791
3888
8984 4080 4C
1 4177
4273
4369
4465
4562
4658
4754
4850
4946 5042 ;
2 5188
5235
5331
5427
5528
5619
5716
5810
5906 6002 !
8 6098
6194
6290
6386
6482
6577
6673
6769
6864 6960
4 7066
7152
7247
7843
7438
7534
7629
7725
7820 , 7916 .
5 8011
8107
8202
8298
8893
8488
8684
8679
8774 ' 8870 
6 8965
9060
9155
9250
9346
9441
9686
9631
9726 9821
7; 9916
0011
0106
0201
0296
0391
0486
0581
0676 0771 !
8 660865
0960
1055
1160
1245
1339
1434
1629
1628 1718
9
1813
1907
2002
2096
2191
2286
2380
2476
2669 2663
1
2
8
4
6
6
7
8 < 9
460
2758
2852
2947
8041
3135
3230
3324
3418
8612 3607 m
1
3701
3795
3889
8983
4078
4172
4266
4860
4464 4648
2
4642
4736
4830
4924
5018
6112
5206
5299
6893 6487
3
5581
5675
5769
5862
5966
6060
6143
6237
6331 I 6424
4
6518
6612
6705
6799
6892
6986
7079
7173
7266 7860
5
7458
7546
7640
7733
7826
7920
8018
8106
8199 8298
6
8886
8479
8572
8665
8759
8862
8945
9088
9131 9224 •
7
9317
9410
9503
9596
9689
9782
9875
9967
0060 : 0153
0988 ■ 1080 !
8 670246
0839
0431
0524
0617
0710
0802
0895
9
1173
1265
1358
1461
1543
1686
1728
1821
1913 , 2005
1
2
8
4
5
6
7
8 1 9
470
2098
2190
2283
2875
2467
2560
2652
2744
2836 2929 4:
1
3021
3118
3205
8297
3390
8482
3674
3666
3758 3850 1
2
3942
4034
4126
4218
4810
4402
4494
4586
4677 , 4769 '
3
4861
4958
5045
5137
5228
5320
5412
5503
5695 1 5687 j
4
5778
5870
5962
6053
6145
6236
6328
6419
6611 1 6602 1
5
6694
6785
6876
6968
7059
7161
7242
7338
7424 ! 7516 '
6 7607
7698
7789
7881
7972
8068
8154
8245
8336
8427,
7! 8518
8609
8700
8791
8882
8978
9064
9155
9246
9357
8! 9428
9519
9610
9700
9791
9882
9973
9680336
0426
0517
0607
0698
0789
0879
0068
0970
0154 0246
1060 1151 ; !
1
1
2
3
4
5
6
7
8
9 i
480 1241
1832
1422
1513
1603
1693
1784
1874
1964
2055 48>
1 2145
2285
2826
2416
2506
2696
2686
2777
2867
2957
2 3047
3137
8227
3317
3407
3497
3587
3677
8767
3857
3 3947
4037
4127
4217
4307
4396
4486
4576
4666
4756
4: 4845
4935
5025
5114
5204
6294
5383
6478
5563
5652:
5 5742
5881
5921
6010
6100
6189
6279
6368
6458
6547 1
6 6686
6726
6815
6904
6994
7083
7172
7261
7351
7440;
7i 7529
7618
7707
7796
7886
7975
8064
8153
8242 8831 1
8, 8420
8509
8598
8687
8776
8865
8953
9042
9181
9220'
9
9809
9898
9486
9575
9664
9753
9841
9930
J
1
2
8
4
5
6
7
0019
8
0107
9
490 690196
0285
0873
0462
0550
0639
0728
0816
0905
0993
49
1' 1081
1170
1258
1847
1485
1524
1612
1700
1789
1877
2 1965
2053
2142
2230
2318
2406
2494
2588
2671
2759
3 2847
2935
3023
3111
3199
3287
3375
8468
3551
3639
4 8727
3815
3903
8991
4078
4166
4254
4842
4430
4517
5 4605
4693
4781
4868
4956
5044
5131
5219
5307
5394
6 5482
5569
5657
5744
5832
5919
6007
6094
6182
6269
7 6356
6444
6581
6618
6706
6798
6880
6968
7055
7141
8 7229
7817
7404
7491
7578
7665
7752
7889
7926
8014
1
9 8IOO: 8188
8275
8362
8449
8585
8622
8709
A7M
8888 1
N. 6000L.
698970]
TABLE 111.
[No. 5499 L. 740284
•1
500
1
1
2
3
4
5
6
7
8
9
N.
698970
9838
9057
9924
9144
9231
9817
9404
9491
9578
9664
9751
500
1
2
0011
0877
0098
0968
0184
1050
0271
1136
0358
1222
0444
1309
0531
1395
0617
1482
2 700704
0790
3 1568
1654
1741
1827
1913
1999
2086
2172
2258
2344
3
4 2431
2517
2603
2689
2776
2861
2947
3033
3119
3205
4
5 8291
3877
8463
3549
3635
3721
3807
3893
3979
4065
6
6 4151
4236
4822
4408
4494
4579
4665
4751
4837
4922
6
7 5008
5094
5179
5265
5350
5436
5522
5607
5693
6778
7
8 5864
5949
6035
6120
6206
6291
6376
6462
6547
6632
8
9; 6718
6803
6888
6974
7059
7144
7229
7315
7400
7485
9
1
1
2
3
4
5
6
7
8
9
510 7570
7655
7740
7826
7911
7996
8081
8166
8251
8386
510
1 8421
8506
9355
8591
9440
8676
9524
8761
9609
8846
8931
9015
9100
9185
1
2
3
9 AIM
9694
9779
9863
9948
0033
0879
2
3
710117
0202
0287
0371
0456
0540
0625
0710
0794
4 0963
1048
1132
1217
1301
1885
1470
1554
1639
1728
4
5 1807
1892
1976
2060
2144
2229
2313
2397
2481
2566
6
6 2650
2734
2818
2902
2986
3070
3154
3238
3323
3407
6
, 7 3491
3575
3659
8742
3826
3910
3994
4078
4162
4246
7
1 8 4330
4414
4497
4581
4665
4749
4833
4916
5000
6084
8
9 5167
5251
5335
5418
5502
5586
5669
5753
5836
5920
9
1 i
1
2
3
4
5
6
7
8
9
'520 6003
6087
6170
6254
6337
6421
6504
6588
6671
6754
620
1 6838
6921
7004
7088
7171
7254
7338
7421
7504
7587
1
2 7671
7754
7837
7920
8003
8086
8169
8253
8336
8419
2
3. 8502
A nooi
8585
9414
8668
9497
8751
9580
8884
9663
8917
9000
9083
9165
9248
3
4
5
voox
9745
9828
9911
9994
0077
0903
4
5
720159
0242
0325
0407
0490
0573
0655
0738
0821
6 0986
1068
1151
1233
1316
1898
1481
1563
1646
1728
6
7 1811
1893
1975
2058
2140
2222
2305
2387
2469
2552
7
8 2634
2716
2798
2881
2963
8045
3127
3209
3291
3374
8
9 3456
3538
3620
8702
3784
3866
3948
4030
4112
4194
9
I '
1
2
3
4
5
6
7
8
9
530 4276
4358
4440
4522
4604
4685
4767
4849
4981
6013
680
/ !< 5095
5176
5258
5340
5422
5503
5585
5667
5748
5830
1
. 2 5912
/ 81 6727
5993
6075
6156
6238
6320
6401
6483
6564
6646
2
6809
6890
6972
7053
7134
7216
7297
7379
7460
3
41 7541
7628
7704
7786
7866
7948
8029
8110
8191
8278
4
6. 8354
8485
8516
8597
8678
8759
8841
8922
9003
9084
6
6 9165
9246
9327
9408
9489
9570
9651
9732
9813
9893
6
7
8
7
8
vv/«
0055
0868
0136
0944
0217
1024
0298
1106
0378
1186
0459
1266
0540
1347
0621
1428
0702
1508
780782
9, 1589
1669
1750
1830
1911
1991
2072
2152
2233
2313
9
^ 1
640 28M
1
2
3
4
5
6
7
8
9
2474
2555
2685
2715
2796
2876
2956
3037
3117
640
1; 8197
8278
8858
3488
3518
3598
3679
3759
3839
3919
1
2
8999
4079
4160
4240
4820
4400
4480
4560
4640
4720
2
'
4800
4880
4960
5040
5120
5199
5279
5359
5439
5519
3
\ *
5599
6679
5759
5S38
5918
5993
6078
6157
6237
6317
4
5
6897
6476
6556
6635
6715
6795
6874
6954
7034
7113
6
\ 2
7198
7272
7852
7431
7511
7590
7670
7749
7829
7908
6
I
7987
8067
8146
8225
8305
8384
8468
8543
8622
8701
7
\i
8781
9572
8860
9661
8939
9731
9018
9810
9097
9177
9256
9386
9414
9493
8
9889
9968
0047
0126
0205 1 0284
9
10
TA5L1 IIL
[ICo. 6999 L. 778079!
8
9 N.
Mai » t*!^ ssti :^.L. .n*.*.
* I ! ri. l^i! l^iir !!;<*
1 "! s?!.* i' '. * ? Vf il"r
3 STiH i>'§ ir*?i iKT
♦ *j: •.;=• 7.^ frii
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f :. : 5:r.? 52?: «:•>
" !.•;.: ;:*•?:? f» :: */•■*>
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> r*:i T**^ r.v~ r*«;
1 ?iH3 >■*•
•■?*?
>'.i*
>*>!
'IT
4
1»44
4f.fl
*i45
5
l<*i4
2411
SIM
4r«2
«323
TlOl
T?:3
6
S653
9427
0915
1703
2439
3275
4053
4S40
5«21
6401
7179
7955
8731
9504
0994 i
17S2
2568 .
3353
4136
4919
5699
6479
7256
8033
1073 .560,
?^^
w?A 0971
1$M 1741
1048 1125
1813 18^5
t^vi 2433 25<.'9 25S6 2663
1860
2647
3431
4215
4997
5777 , 6
S556 ' 7
7384 ; 8
8110 ; 9
8 ' 9 : '
SS08 8885 560
9582 9659 1
.»45 0:23 02i>0 0277 0354 0431
j*^,"*;*? ;:« y.i 4..
4 \ir'r :«^ :*.*.; \s'.'j
I r:4* 1115 iri r*r*
« !:•:•? i>i^ *r.; ^;4T 3:23 3>» 3277 3353 8430 3506
7 4^<! }.r.> i'M 3.*: 3
* 4M? 44i5 4J : 457*
1202
1972
2740
3^!^ 3v^ 4<:42 4119 4195 4272
46J4 473*) 4><>7 4883 4960 5036
9
"Hi
£:i*
5it5
5341
5417
5494
5570
5646
5722
6799
9
V
:
.
3
4
5
6
7
8
9
570
:.*::
:Vi;
t>:7
fi.'VS
6:j;i
6256
6.^2
6403
6484
6560 570
^
A
e^
■>::2
?"ji
6>o4
6:='40
7«>16
7092
7163
7244
7820
o
r;^.
7472
754?
7624
77C1
7775
7S51
7927
8008
6079
3
s::<
*i:?^j
S.^>o
5&S2
84^8
S533
wk»9
8685
8761
8836
4
s^:i
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sa
^•139
y2l4
92y<i
9366
9441
9517
9592
5
««*
t743
9?1>
9S94
9k'70
0799
0121
0S75
0196
0950
0272
1025
0347
1101
! 6:
6'>422
•>49S
0573
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0724
7
ri:«5
1251
132^
1402
1477
1552
1627
1702
1778
1858
: 8
i**j*
2.i«>3
2»»75
2153
222S
23'.'3
2378
2453
2529
2604
9
2679
2754
2529
29i'4
2978
3053
3128
3203
3278
3853
1
2
3
4
5
6
7
8
9
580
8428
3503
3578
3653
8727
3802
3877
8952
4027
4101 5S0
' 1
4176
4251
4326
4400
4475
4550
4624
4699
4774
4848
1
. 2
4923
4998
5*172
5147
5221
5296
5370
5445
5520
5594
2
• 8
5669
5743
5S13
5892
5966
6041
6115
6190
6264
6838
8
4
6413
6487
6562
6636
6710
6785
6859
6933
7007
7082
4
, 5
7156
7230
73rt4
7379
7453
7527
7601
7675
7749
7828
5
i 6
7898
7972
8046
8120
8194
8268
8342
8416
8490
8564
6
7
8638
8712
8786
8860
8934
9008
9082
9156
9230
9803
(
8
9377
9451
9525
9699
9673
9746
9820
9894
9968
0042 r
8
9 770116
0189
0263
0336
0410
0484
0557
0631
0705
0778
9
;
1
2
3
4
5
6
7
8
9
690
0852
0926
0999
1073
1146
1220
1298
1867
1440
1514
590
1
1587
1661
1734
1808
1881
1955
2028
2102
2175
2248
1
•2
2
2322
2395
2468
2542
2615
2688
2762
2885
2908
8981
8
8055
3128
3201
8274
3348
8421
3494
8567
3640
3718
S
4;
8786,
3860
8933
4006
4079
4153
4225
4298
4371
4444
4
6.
46171
4590
4663
4786
4809
4882
4955
5028
5100
5178
5
6
5246,
5819
5892
5465
5538
5610
5683
5756
5829
5902
6
5
5974 1
6047
6120
6193
6265
6838
6411
6488
6556
66S9
7,
6701
6774
6846
6919
6992
7064
7137
7209
7888
7354
8
ol
7427
7499
7572 i 7644
7717 1 7789
7862
7984
8006 1 807»
9
11
Hi. 6000 L. 778161]
TABLE III.
[No. 6499 L. 812847
N.
1
2
8
4
5
6
7
8
9
N.
600
600
778161
8224
8296
8868
8441
8513
8686
8668
8780
8802
1
8874
8947
9019
9091
9163
9286
9308
9380
9452
9624
1
2
9096
9669
9741
9818
9885
9957
0029
0749
0101
0821
0173
0893
8
780817 0889
0461
0538
0605
0677
9246
0965
2
8
4
1087
1109
1181
1253
1324
1896
1468
1640
1612^
1684
4
5
1755
1827
1899
1971
2042
2114
2186
2268
2329
2401
5
6
7
2473
2544
2616
2688
2759
2831
2902
2974
3046
8117
6
8189 8260 8882
8408
3475
3546
3618
8689
3761
8832
7
8
8904 8975 . 4046
4118
4189
4261
4382
4408
4475
4646
8
9
4617 4689 1 4760
4831
4902
4974
6046
6116
6187
6269
9
6 1,2
3
4
5
6
7
8
9
610
5880 5401 ! 5472
5543
5615
5686
6767
5828
5899
5970
610
1
60411 6112 6188
6254
6325
6396
6467
6538
6609
6680
1
2
67511 6822 ; 6893
6964
7035
7106
7177
7248
7319
7390
2
8
7460, 7581 7602
7673
7744
7815
7885
7966
8027
8098
8
4
8168'
8289 ! 8310
8381
8451
8522
8693
8663
8734
8804
4
6
8875
8946 9016
9087
9167
9228
9299
9369
9440
9610
5
6
9581
9651
9722
9792
9863
9933
0004
0707
0074
0778
0144
0848
7
790285
0856
0426
0496
0567
0637
0216
0918
6
7
8
0988
1059
1129
1199
1269
1340
1410
1480
1560
1620
8
9
1691
1761
1881
1901
1971
2041
2111
2181
2262
2322
9
1
2
3
4
5
6
7
8
9
620
2892
2462
2532
2602
2672
2742
2812
2882
2962
8022
620
1
' 8092 8168
8231
3301
3371
3441
3611
8581
8661
3721
1
2
8790
8860 3980
4000
4070
4139
4209
4279
4349
4418
2
8
4488
4558 ! 4627
4697
4767
4836
4906
4976
6045
6115
8
4
5185
5254
5324
5398
5468
5632
6602
6672
6741
5811
4
5
5880
5949
6019
6088
6158
6227
6297
6366
6486
6506
5
6
6574
6644
6713
6782
6852
6921
6990
7060
7129
7196
6
7
7268 7887
7406
7475
7545
7614
7683
7762
7821
7890
7
8
7960
8029
8098
8167
8286
8805
8874
8443
8618
8582
8
9
8651
8720
8789
8858
8927
8996
9065
9134
9203
9272
9
1
2
3
4
5
6
7
8
9
10
9841
9409
9478
9547
9616
9686
9764
9823
9892
9961
630
1
I
800029
0098
0167
0236
0306
0373
0442
0611
0680
0646
2
0717
0786
0854
0923
0998
1061
1129
1198
1266
1885
2
\
, 1404. 1472
1541
1609
1678
1747
1815
1884
1952
2021
3
2089
2158
2226
2295
2363
2432
2600
2668
2637
2706
4
•774
2842
2910
2979
3047
3116
3184
3262
3321
3389
5
8457
8525
8594
8662
3730
3798
8b67
3935
4003
4071
6
4189
4208
4276
4844
4412
4480
4548
4616
4685
4753
7
4821
4869
4967
5025
5093
5161
5229
6297
5365
5438
8
5501
5569
5637
5705
6778
6841
5908
5976
6044
6112
9
1
2
8
4
5
6
7
8
9
6180
6248
6816
6384
6451
6619
6587
6655
6728
6790
640
6858
6926
6994
7061
7129
7197
7264
7832
7400
7467
1
7585
7603
7670
7788
7806
7878
7941
8008
8076
8148
2
8211
8279
8346
8414
8481
8649
8616
8684
8761
8818
3
8886
8958
9021
9088
9156
9223
9290
9368
9425
9492
4
9560
9627
9694
9762
9829
9896
9964
0031
0703
0098
0770
>2S8
0800
0367
0434
0501
0669
0636
0165
0837
5
6
»904
0971
1039
1106
1178
1240
1807
1374
1441
1508
7
575
1642
1709
1776
1848
1910
1977
2044
2111
2178
8
M5
2312 2379
2445
2512
2579
2646
2718
2780
2847
9
12
1359; 1422
1985 2047
2609 2672
3283 3295
8855 3918
4477 4589
0232
0859
1485
2110
, 2734
I 8857
8980
4601
13
No.
N.
700
7000 L.
845098]
TABLE III.
[No. 7499 L, 876003
1
2
3
4
5
6
7
8
9
N.
845098
5160
^ 5222
5284
5346
5408
5470
5532
5594
5656
700
1
5718
5780
5842
5904
5966
6028
6090
6151
6213
6275
1
2
6337
6399
6461
6523
6585
6646
6708
6770
6832
6894
2
3
6955
7017
7079
7141
7202
7264
7826
7388
7449
7511
8
4
7573
7634
7696
7758
7819
7881
7943
8004
8066
8128
4
6
8189
8251
8312
8374
8435
8497
8559
8620
8682
8743
5
8805
8866
8928
8989
9051
9112
9174
9235
9297
9358
6
7
8
9419
9481
9542
9604
9665
9726
9788
0401
9849
9911
9972
7
8
850083
0095
0156
0217
0279
0340
0462
0524
0585
9
0646
0707
0769
0830
0891
0952
1014
1075
1136
1197
9
1
2
3
4
6
6
7
8
9
710
1258
1320
1381
1442
1508
1564
1625
1686
1747
1809
710
1
1870
1981
1992
2058
2114
2175
2236
2297
2358
2419
1
2
2480
2541
2602
2663
2724
2785
2846
2907
2968
3029
2
8
8090
8150
8211
3272
8333
3394
3455
3516
8577
3637
8
4
8698
3759
3820
3881
3941
4002
4063
4124
4185
4245
4
6
4306
4867
4428
4488
4549
4610
4670
4731
4792
4852
5
6
4913
4974
5034
5095
5156
5216
5277
5337
5898
5459
5
7
5519
5580
5640
5701
5761
5822
5882
5943
6003
6064
7
8
6124
6185
6245
6306
6366
6427
6487
6548
6608
6668
8
9
6729
6789
6850
6910
6970
7081
7091
7152
7212
7272
9
1
1
2
8
4
5
6
7
8
9
720
7382
7893
7458
7518
7574
7684
7694
7755
7815
7875
720
1
7935
7995
8056
8116
8176
8236
8297
8357
8417
8477
1
2
8537
8597
8657
8718
8778
8838
8898
8958
9018
9078
2
8
9138
9198
9258
9318
9379
9439
9499
9559
9619
9679
8
4
9739
9799
9859
9918
9978
0038
0637
0098
0697
0158
0757
0218
0817
0278
0877
4
5
5
860838
0398
0458
0518
0578
6
0987
0996
1056
1116
1176
1236
1295
1355
1415
1475
6
7
1534
1594
1654
1714
1773
1833
1893
1952
2012
2072
7
8
2131
2191
2251
2310
2370
2430
2489
2549
2608
2668
8
9
2728
2787
2847
2906
2966
3025
3085
3144
3204
3268
9
1
2
3
4
5
6
7
8
9
780
8828
8382
3442
3501
3561
3620
3680
3739
3799
3858
780
1
8917
8977
4036
4096
4155
4214
4274
4833
4892
4452
1
2
4511
4570
4630
4689
4748
4808
4867
4926
4985
5045
2
8
6104
5168
5222
5282
5341
5400
5459
5519
6578
6687
3
4
5696
5755
5814
5874
5933
5992
6051
6110
6169
6228
4
5
6287
6846
6405
6465
6524
6583
6642
6701
6760
6819
6:
6
6878
6937
6996
7055
7114
7173
7282
7291
7350
7409
6
7
7467
7526
7585
7644
7703
7762
7821
7880
7939
7998
7
8
8056
8115
8174
8233
8292
8350
8409
8468
8527
8586
8
9
8644
8708
8762
8821
8879
8938
8997
9056
9114
9173
9
1
2
3
4
5
6
7
8
9
740
9282
9290
9349
9408
9466
9525
9584
9642
9701
9760
740
1
9818
9877
9935
9994
0053
0638
0111
0696
0170
0755
0228
0813
0287
0872
0345
0930
1
2
2
870404
0462
0521
0579
8
0989
1047
1106
1164
1223
1281
1339
1898
1456
1516
8
4
1573
1631
1690
1748
1806
1865
1923
1981
2040
2098
4
6
2156
2215
2273
2331
2389
2448
2506
2564
2622
2681
5
6
2739
2797
2855
2913
2972
3030
3088
8146
3204
3262
6
7
8821
8879
8487
8495
3553
3611
3669
3727
8785
8844
7
8
8902
8960
4018
4076
4134
4192
4250
4308
4866
4424
8
9
4482
4540
4598
4656
4714
4772
4880
4888
4945
5008 9
14
Ve. 7M0 L. 875061] TABLB III.
[Ifo. 7000 L. 0O8O86
750
1
2
%
4
5
6
7
8
875061
5119
5177 1 6285
5298
5851
5400
6466
5524
5582 ,750
1
5640 5698
5756 581S
5871
6980
5087
6045
6108
6160 1 1
2
6218 6276
6338 1 6891
6449
6507
6564
6622
6680
6787 1 2
8
6795 6853
6910 : 6968 7026
7088
7141
7100
7266 7814 ' 3
4
7871 7429
7487 7544 ■ 7602
7659
7717
7774
7882
7889 , 4i
5
7947 8004
8062 8110 j 8177
8284
8202
8840
8407
8464 5'
6
8522 8579
8637 < 8694 > 8752
8809
8866
8024
8981
9039 1 6
7
9096 9158
0211 0268 1 0825
9888
0440
0407
9565 9612 7
8
9669 9726
0784
9841
9898
9956
1
0018
0070
0127 ' 0185 8
9
880242 0299
0856 0418
0471
0528
0585
0642
0600 J 0756 9
1
2 8
4
6
6
7
8 , • '
760
0814 0871
0028 0085
1042
1099
1156
1218
1271 , 1828 760
1
1885 1442
1499 1556
1613
1670
1727
1784
1841 * 1898 1
2
1955 2012
2069 2126
2183
2240
2297
2854
2411 1 2468 ' 2
8
2525 2581
2<J38 ; 2695  2752
2809
2866
2028
2980 80S7 3
4
8093 8150
8207 8264 ; 8321
3377
8484
8491
8548 8605 4
5
8661 3718
8775 8882 i 8888
8945
4002
4059
4115 41T2 > 5
6
4229 4285
4842 4899 ; 4455
4512
4569
4625
4682 47S9 6
7
4795 4852
4909 ! 4965 i 5022
5078
5135
5192
5248 1 5805 7
8
5861 5418
5474 i 5581 5587
5644
5700
5757
5818 5870 8
5926 5983
6039 6096
6152
6209
6265
6821
6878
6434 9
1
2 1 8
4
5
6
7
8
t
770
6491 6547
6604 6660
6716
6778
6829
6885
6942
6908 770
1
7064 7111
7167 1 7223
7280
7336
7392
7449
7505
7561 1
2
7617 7674
7730 '■ 7786
7842
7898
7955
8011
8067
8123 2
8
8179 8236
8292 ' 8348
8404
8460
8516
8578
8629
8685 ' 8
4
8741 8797
8858 ; 8909
8965
9021
9077
9184
9190 9246 i
5
9302 9358
9414 9470
9526
9582
9638
9694
9750 9806 . 5
9862 9918
1
9974
0086
0645
0141
0700
0197
0756
0253
0812
0309 ; 0365 «
0868 I 0924 7
7
890421 0477
0533 0589
8
0980 1035
1091 1147
1208
1259
1314
1370
1426 1482 8
1537
1593
1649 1705
1760
1816
1872
1928
1083
2039 r
1
2 i 8
4
5
6
7
8
1 '
780
2095 2150
2206 i 2262
2317
2873
2429
2484
2540
2505 T80
1
2651 2707
2762 i 2818
2873
2929
2985
8040
8006 8151 1,
2
3207 8262
3818 ': 8373
3429
8484
8540
8505
3651
8706. J
8
8762 3817
3878 8928
8984
4089
4094
4150
4205
4861 S
4
4316 4371
4427 ; 4482
4538
4593
4648
4704
4759
4814 <
5
4870; 4925
4980 5086
5091
5146
5201
5257
5812 5867 \
6
5423 5478
5533 i 5588
5644
5699
5754
5800
5864
5920, •
7
5975
6080
6085 : 6140
6195
6251
6806
6861
6416
6471 '
8
6526
6581
6636
6692
6747
6802
6857
6912
6067
7022 8
7077
7182
7187
7242
7297
7852
7407
7462
7517
7572
»,
1
2
3
4
5
6
7
8
•
700
7627
7682
7787 ; 7792
7847
7902
7957
8012
8067
8122 ITW!
8670 1
1
8176
8281
8286
8341
8396
8451
8506
8561
8615
e
8725
8780
8885
8890
8944
8999
9054
9109
9164
0218
8
0278
0828
9863
9487
9492
0547
9602
0656
0711
9766
4
0821
9875
9980
9985
0089
0586
0004
0640
0149
0695
0808
0740
0258
0804
0312
0850
5
000867
0422
0476
0581
a
0918
0068
1022
1077
1181
1186
1240
1805
1840
1404
r
1458
1518
1567
1682
1676
1781
1785
1840
1804
1048
8
2008
2057
2112
8166
8221
2275
2829
2884
2488
2198
2647
2601
2655
8710
2764
8818
8878
2027
2081
8686
__\
15
N«.
8000 L. 908090]
TABLB III.
[ir«. 8499 L. 9298n!
N.
1
2
8
4
6
6
r
8
9
H.
800
908090
8144
8199
8258
8307
3861
3416
8470
8524
8578
800
1
8888
8687
8741
8795
3849
3904
3958
4012
4066
4120
1
2
4174
4229
4288
4887
4391
4445
4499
4663
4607
4661
2
8
4716
4770
4824
4678
4932
4986
5040
5094
5148
6202
4
5256
6810
5864
5418
5472
5526
5580
5684
5688
5742
6
5796
5850
5904
5958
6012
6066
6119
6173
6227
6281
6
6885
.6889
6448
6497
6551
6604
6658
6712
6766
6820
7
6874
6927
6981
7086
7089
7148
7196
7250
7804
7868
8
7411
7465
7519
7578
7626
7680
7784
7787
7841
7895
8
9
7949
8002
8056
8109
8163
8217
8270
8824
8878
8431
9
1
2
8
4
5
6
7
8
9
.810
8485 8589
8592
8646
8699
8768
8807
8860
8914
8967 810
1
9021 9074
9128
9181
9285
9289
9842
9896
9449
9603 ' ll
2
8
9556 9609
9663
9716
9770
9828
0858
98n
9980
OOfil
0037
0571
2
8
910091 0144
0197
0251
0804
0411
0464 0618
4
0624 0678
0781
0784
0888
0891
0944
0998 1051
1104
4
' 6
1158 1211
1264
1817
1871
1424
1477
1630 1584
1687
5
6
1690 1743
1797
1850
1903
1956
2009
2063 • 2116
2169
6
7
2222
2276
2828
2881
2485
2488
2541
2694 2647
2700
7
8
2758
2806
2859
2918
2966
8019
8072
8126 3178
8281
8
9
8284
8887
8890
8443
8496
8549
8602
8666 8708
8761
9
1
2
8
4
5
6
7
8
9
820
8814
8867
8920
3973
4026
4079
4132
4184
4287
4290
820
1
4848 4896
4449
4502
4555
4608
4660
4713
4766
4819
1
2
4872
4925
4977
5030
5083
5186
5189
6241
6294
5347
2
S
5400
5458
5505
5558
5611
5664
6716
5769
5822
5875
3
4
5927
5980
6083
6085
6188
6191
6248
6296
6349
6401
4
6
6454, 6507
6559
6612
6664
6717
6770
6822
6875
6927
5
6
6980 7088
7085
7138
7190
7248
7295
7348
7400
7 last
({
7
7506 7558
7611
7663
7716
7768
7820
7873
7926 7978
7
a
8080
8088
8185
8188
8240
8293
8846
8397
8450 ' 8602
8
9
8555
8607
8659
8712
8764
8816
8869
8921
8978 9026
9
1
2
8
4
5
6
7
A
g
830
9078
9180
9188
9285
9287
9340
9392
9444 9496
9549
880
1
2
^601
9658
9706
9758
9810
9862
9914
9967
1
2
0019
0541
0071
0698
920128 0176
0228
0280
0332
0384
0436
0489
S
0645; 0697
0749
0801
0853
0906
0968
1010 1062
1114
8
4
11661 1218
1270
1322
1374
1426
1478
1630
1582
1634
4
6
1686 1788
1790
1842
1894
1946
1998
2060
2102
2154
5
6
2206. 2258
2810
2362
2414
2466
2618
2670
2622
2674
5
7
27251 2777
2829
2881
2983
2985
8087
8089
8140
8192
7
8
8
3244: 8296
8848
8399
8451
8508
8665
8607
8668
8710
9
8762, 8814
8865
8917
3969
4021
4072
4124
4176
4228
9
1
2
8
4
5
6
7
8
9
4744
840
4279
4881
4888
4434
4486
4588
4689
4641
4698
840
1
4796 4848
4899
4951
5003
5054
5106
6157
6209
5261
1
2
2
5812 5864
5415
5467
5518
5570
5621
5678 5725
5776
8
5%2» 5879
5981
5982
6084
6085
6187
6188 6289
6291
8
4
5
6
7
8
9
4
6842 6894
6445
6497
6548
6600
6651
6702 ; 6764
* 6805
6
6857
6908
6959
7011
7062
7114
7166
7216 i 7268
7819
7832
8
7870
7422
7478
7524
7576
7627
7678
7780 1 7781
7
7888
7985
7986
8087
8088
8140
8191
8242
8298
8845
8
8896
8447
8498
8549
8601
8652
8708
8754
8805
8857
9
8908
8959
9010
9061
9112
9163
9S15
9266
19817
9868
16
,No. S5iiO L. 929419]
TABLE III.
[No. 8999 L. 954194
1240 , 1289 ?i
1726 ' 1776 {
2211 . 2259 5
2696 I 2744 <
3180 I 3228 : ,
866S i 8711 ^i
4146 I 4194 I '
17
5«. 9000 L. W42i«\
TABLB XXL
Pf 0. 9499 L. 977678
K.
1
2
8
4
5
6
7
8
9
N.
990
984248
4291
4339
4387
4436
4484
4582
4580
4628
4677
900
1
4726
4778
4821
4869
4918
4966
5014
5062
5110
5168
1
2
8207
5288
8308
5352
5399
5447
5495
5543
5692
5640
2
3
8888
5736
5784
5882
5880
5928
5976
6024
6072
6120
3
4
8188
6216
6265
6313
6861
6409
6457
6505
6653
6601
4
5
8649
6697
6745
6798
6840
6888
6936
6984
7032
7080
5
6
7128
7176
7224
7272
7320
7868
7416
7464
7612
7559
6
7
7807
7688
7703
7751
7799
7847
7894
7942
7990
8038
7
8
8086
8184
8181
8229
8277
8825
8373
8421
8468
8516
Sj
9
8864
8612
8659
8707
8755
8803
8850
8898
8946
8994
9
1
2
3
4
6
6
7
8
9
910
9041
9089
9137
9185
9282
9280
9328
9375
9423
9471
910
1
9818
9586
9614
9861
9709
9757
9804
9852
9900
9947
1
^i
•fifl/k
wvo
0042
0518
0090
0566
0138
0613
0185
0661
•233
0709
0281
0766
0328
0804
6376
0851
0428
0899
2
3
8
980471
4
0946
0994
1041
1089
1136
1184
1281
1279
1326
1874
4
6
1421
1489
1816
1863
1611
1658
1706
1753
1801
1848
5
1895
1943
1990
2038
2085
2132
2180
2227
2275
2322
6
7
2889
2417
2464
2511
2559
2606
2663
2701
2748
2796
7
8
2848
2890
2987
2988
3032
8079
8126
3174
8221
3268
8
9
8816
3863
3410
3457
8504
8552
8599
3646
3693
3741
9
1
2
3
4
5
6
7
8
9
ftIO
8788
3838
3882
3929
3977
4024
4071
4118
4165
4212
920
1
4260
4807
4854
4401
4448
4495
4542
4690
4637
4684
1
2
4781
4778
4825
4872
4919
4966
5018
5060
5108
6155
2
8
8202
5249
5298
5343
5890
5487
6484
5631
6678
5625
8
4
8872
5719
5766
5813
5860
6907
5954
6001
6048
6096
4
5
8142
8189
6286
6283
6829
6376
6428
6470
6617
6664
6
8
6611
6658
6705
6752
6799
6845
6892
6939
6986
7083
6
7
7080
7127
7178
7220
7267
7314
7361
7408
7464
7601
7
8
7848
7595
7642
7688
7736
7782
7829
7876
7922
7969
8
9
8016
8062
8109
8156
8203
8249
8296
8348
8389
8486
9
1
2
8
4
6
6
7
8
9
980
8483
8580
8576
8628
8670
8716
8763
8810
8866
8903
930
1
8980
8998
9048
9090
9186
9183
9229
9276
9823
9869
1
2
9416
9463
9509
9556
9602
9649
9695
9742
9789
9836
2
8
9882
9928
9975
0021
0486
0068
0588
0114
0579
0161
0626
0207
0672
0254
0719
0800
0765
3
4
4
970847
0398
0440
6
0812
0858
0904
0951
0097
1044
1090
1187
1188
1229
5
6
12T6
1822
1869
1415
1461
1608
1564
1601
1647
1698
6
7
1740
1786
1832
1879
1925
1971
2018
2064
2110
2157
7
(
2203
2249
2295
2842
2888
2434
2481
2527
2678
2619
8
fl
2666
2712
2758
2804
2851
2897
2943
2989
8085
8082
9
1
2
8
4
5
6
7
8
9
I4C
8128
3174
8220
3266
3813
8869
3405
3451
8497
8543
940
1
8880
3636
8682
3728
3774
8820
8866
3913
8969
4005
1
3
4081
4097
4148
4189
4286
4281
4327
4374
4420
4466
2
2
4812
4858
4604
4650
4696
4742
4788
4834
4880
4926
3
4
4972
8018
8064
5110
5156
5202
5248
5294
5840
5886
4
e
8482
8478
8524
5570
5616
5662
5707
5753
5799
5846
6
fl
8891
8987
8988
6029
6075
6121
6167
6212
6258
6304
6
7
8880
6396
6442
6488
6538
6579
6625
6671
6717
6763
7
8
MOO
8854
6900
6946
6992
7037
7083
7129
7175
7920
8
9
7886
7812
7858
7408
7449
7495
7641
7586
7632
7678
9
18
No. 9600 L. 977724]
TABLE III.
[No. 9999 L. 999987!
N.
1
2
8
4
6
6
7
8
9 N.
050 977724
7769
7816
7861
7906
7962
7998
8043
8089
8185 950
1
8181
8226
8272
8817
8863
8409
8464
8600
8546
8591 1
2
8637
8683
8728
8774 8819
8865
8911
8956
9002
9047 S
3
9093
9138
9184
9230 9276
9321
9366
9412
9457
9503 8
4
5
9548
9594
9639
9686 1 9730
9776
9821
9867
9912
9958 4
980003
0049
0094
0140 ' 0186
0281
0276
0822
0867
0412 6'
6
0458
0503
0549
0594
0640
0685
0730
0776
0821
0867 6,
7
0912
0957
1008
1048
1093
1189
1184
1229
1275
1820 7l
8
1366
1411
1456
1501
1647
1692
1637
1688
1728
1778 1 8
9
1819
1864
1909
1954 1 2000
1
2045
2090
2185
2181
2226 9
1
1
2
8
4
6
6
7
8
• 1
960
22n
2816
2362
2407
2462
2497
2643
2588
2688
2678 960
8180 ll
1
2728
2769
2814
2859
2904
2949
2994
8040 ■ 8085
2
8175
3220
8266
8310
8856
8401
8446
8491 ' 3586
8581 1 Si
8
8626
3671
8716
3762 i 8807
8852
8897
8942 , 8987
4082 S
4
4077
4122
4167
4212 . 4257
4802
4847
4892 1 4487
4482 4i
5
4527
4572
4617
4662
4707
4752
4797
4842 4887
4982 6;
6
4977
6022
6067
6112
6167
6202
6247
6292 5837
5882 6'
7
5426
6471
6516
6561
6606
6651
6696
6741 5786
5880 1 7i
8
6876
5920
6965
6010
6056
6100
6144
6189 1 6284
6279 1 Si
9
6324
6369
6418
6458
6593
6548
6598
6687 6682
6727 9
1
2
8
4
6
6
7
8
»
970
6772
6817
6861
6906 1 6951
6996
7040
7085
7180
7176 970
1
7219
7264
7309
7863 ! 7398
7448
7488
7682
7577
7622 1
2
7666
7711
7766
7800
7846
7890
7984
7979
8024
8068 i
8
8113
8157
8202
8247
8291
8386
8881
8425 , 8470
8514 i
4
8559
8604
8648
8698
8737
8782
8826
8871 1 8916
8960 4
6
9005
9049
9094
9138 ' 9188
9227
9272
9816 9361
9405 5
6
9450
9494
9639
9688 9628
9672
9717
9761 9806
9850 «
7
9895
9939
9983
0028 0072
0472 1 0616
0117
0561
0161
0605
0206 1 0250
0650 0694
0294 r
0788 8
8
990339
0388
0428
9
0783
0827
0871
0916 0960
1
1004
1049
1098
1187
1182 9
1
2
8
4
6
6
7
8
9
980
1226
1270
1316
1869
1408
1448
1492
1586
1580
1625 980
1
1669
1718
1768
1802
1846
1890
1935
1979
2028
2067 1
2
2111
2166
2200
2244
2288
2888
2877
2421
2465
2509 i
8
2564
2598
2642
2686 2730
2774
2819
2868
2907
2961 8
4
2995
3039
3088
3127 8172
8216
8260
8804
8848
8892 j 4
5
8436
8480
8524
8668 8618
8667
8701
8745
8789
88331 6
6
8877
8921
8966
4009 4068
4097
4141
4185
4229
4878 1 «
7
4817
4361
4406
4449 4498
4687
4681
4625
4669
4718; 7
8
4767
4801
4846
4889 1 4938
4977
6021
5065
5108
5158 1 8
9
6196
6240
6284
6328 ! 6372
6416
6460
5504
5547
5591 9
1
2
1
8 1 4
6
6
7
8
9
990
6685
6679
6728
6767 ! 6811
6864
6898
5942
5986
6080 990
1
6074
6117
6161
6206
6249
6298
6887
6880
6424
6468 1
2
6612
6566
6699
6648
6687
6781
6774
6818
6862
6906 S
8
6949
6998
7037
7080
7124
7168
7212
7255
7299
7848 8
4
7886
7480
7474
7617
7661
7606
7648
7692
7786
7779 4
6
7823
7867
7910
7964
7998
8041
8086
8129
8172
8216 f
6
8269
8808
8847
8890
8484
8477
8621
8564
8608
8652 6
7
8696
8789
8782
8826
8869
8918
8966
9000
9048
9087 T
8
9180
9174
9218
9261
9806
9848
9892
9485
9479
9528 8
9
9666
! 9609
9662
9696
9789
9788
9826
9870
9918
9957 9
10
Deo.
TABLB lY.— Loo. Snris, no.
Sine
86
87
88
89
40
41
48 086965
48 8.097188
44 107167
"' 116926
8 866816 ;"
9 417968 lllll
10 463725 lll^
11 605118 2S2?
12 542906 ^"^^^
18 7.577668 ^J^JJ
14 609858 f3«*J
16 689816 lll^l
16 667846 ^Jj*
17 694178 «;5J
18 7189971 ll^ll
19 7.742477^ ^\^
20 764754! llltL
21
28 825451 2175
24 843984 K
25 7.861662 „„«»
26 878695. *??;
27 895085! Jj^i
28 910879 JSoo
29 926119 *JJS
^ .!tr«o ^^^
81 7.955082 ^
82 968870 ^g
188 982288' *J*JS
!84 995198 *1J2?
85 8.007787j IgSSJ
86 020021! ?2!?i
S.081919
043501
054781
19802
18801
076500 ii?I*
076500
086965
,^ I N "'"— ■ *
45 J.XVV2V
46 126471
47 185810
48 144958
50 162681
171280 j:^
49 8.158907
51 1 # xxov
179718 ^I^SS
» 187985 J^XX
54 196102 \f^
K 8.204070 laJT;
56 211895 J22JJ
57 219581 }!iS
58 227184 {^4
19 284557 igS
«J 2 41855 "^^
17441
17081
16689
16265
15908
15566
15288
14924
14622
14888
COMC.
Infinite.
13.586274
285244
059158
12.984214
837304
758128
12.691176
633184
582082
536275
494882
457094
12.422382
890147
860184
332155
805827
281008
12.257523
235246
214057
193854
174549
156066
12.188338
121805
104915
089121
078881
059158
12.044918
081180
017767
004802
11.992218
979979
11.968081
956499
945219
984224
928500
918035
11.902817
892883
888074
878529
864190
855047
11.846098
887819
828720
820287
812015
808898
11.795980
788105
780419
772866
765448
758145
Tang.
D.
0.000000
6.463726
764756
940847
7.066786
162696
241878
7.808825
866817
417970
463727
505120
542909
7.577672
609857
639820
667849
694179
719003
7.742484
764761
785951
806155
825460
843944
7.861674
878708
895099
910894
926134
940858
7.955100
968889
982253
995219
8.007809
020045
8.031945
048527
054809
065806
076581
086997
8.097217
107202
116963
126510
185851
144996
8.153952
162727
171328
179763
188036
196156
8.204126
211953
219641
227195
284621
241921
601717
293485
208281
161517
131969
111578
96653
85254
76263
68988
62981
57938
58642
49989
46715
43882
41373
39136
37128
85316
33673
32176,
308071
29549
28890
27318
Cotang.
Secant Ootang.
254011
245401
28785
22982
22276
21610
20983
20392
Infinite.
13.636274
285244
069153
12.934214
837304
768122
12.691176
633188
582030
536278
494880
467091
12.422328
390143
860180
832161
805821
280997
12.257516
235239
214049
193845
174640
156066
12.138326
121292
104901
089106
073866
069142
12.044900
081111
017747
004781
11.992191
979966
19888'
198051 ^^•^^^^^'^
J?r!!xl 966473
18808
I8327I
17875
17444
17084
945191
934194
923469
913003
16642; ll»02783
16268,
15912
15568
16241
149271
14625
148861
14067
13790,
18532,
182841
13044
12814J
12590'
12876
12168
D.
892798
883037
873490
864149
865004
11.846048
887273
828672
820287
811964
803844
11.795874
788047
780359
772805
766879
758079
Secant D.l Cosine
10.000000
1
10.000001
1
1
2
2
8
10.000003
4
4
6
5
6
10.000007
7
8
9
10
11
10.000011
12
13
14
15
17
10.000018
19
20
21
28
24
10.000025
27
28
29
81
82
10.000084
36
87
89
41
42
10.000044
46
48
50
52
54
10.000056
58
60
62
64
66
Tang.
CoMC.
10.000000
9.999999
9.999999
9
9
8
8
7
9.999997
6
6
5
5
4
9.999993
3
2
1
89
9.999989
88
87
86
85
83
9.999982
81
80
79
77
76
9.999975
78
72
71
69
68
9.999966
64
68
61
59
58
9.999956
54
52
50
48
46
9.999944
42
40
88
86
84
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
86
85
84
38
82
81
30
29
28
27
26
25
24
28
22
21
20
19
18
17
16
15
14
18
12
11
10
9
8
7
6
5
4
8
2
1
Sine
89 Dto.
20
1 Deg.
TABLE lY.— Loo. 8iffB, im
Sin*
Coflec.
Tang.
Cotang.
,8.241855
1 I 41*033
2 , 56«>y4
63042
69SS1
7t>614
83243
7 . 8.289773
8 96207
9 8.302546
10 . 08794
U. 14954
12 , 21027
18 8.327016
14 , 32924
15.
16
17.
18:
38753
44504
501S1
55783
19 8.361315
20.
21
22!
23 j
24 I
66777
72171
774in»
82762
87962
25 8.393101
26 98179
27.8.403199
28
29
80
08161
13068
17919;
32
31 ; 8.422717
274621
32156:
36800
41394'
45941:
34 I
35 I
36 t ,^...,
87 : 8.450440
88 i 54893
39!
401
42 I
59801
63665]
67985;
72263,
44
45
46
47
48
49
50
51
52
53
54
55
56
,67
i58
59
60
80693
84848
88963
93040
97078
8.501080
05045
08974
12867
16726
20551
8.524843
28102
81828
85528
89186
42819
11963
1176S
115S0
1139S
11221
11050
10883
10723
10565
10413
H.»266
10122
9982
9S47
9714
9586
9460
9338
9219
9103
8990
8880 ;
8772 i
8667 j
8564
8464
8366
8271
81771
8086,
7996;
7909 ;
7823 !
7740
7657
7577
7499
7422
7846
7273
7200
7129
7060
6991
6924
6859
6794
6781
6669
6608
6548
6489
6482
6375
6319
6264
6211
6158
6106
6055
11.758145 8.241921
50967 49102
43906
86958
30119
23386
16757
56165
63115
69956
76691
83323
11.710227
03793
11.697454
91206
85046
78973
11.672984
67076
61247
55496
49819
44217
11.638685
33228
27829
22501
17238
12033
11.606899
01821
11.596801
91839
86932
82081,
11.577283'
72538
67844
63200
58606^
54059
11.549560,
4510'
40699
86335
82015
2773
11.523502
1930!
15152
11037
06960
02922
11.498920
94955
91026
87133
83274
79449
11.47565'
71898
68172
64477
60814
57181
8.289856
96292
8.302634'
08884
15046
21122^
8.827114'
33025
38856
44610.
50289
55895.
8.861430'
66895
722921
77622'
82889!
88092;
8.893234
98315;
8.4033381
08304!
13213J
18068{
8.422869!
27618!
82315
86962
41560 ::
46110
8.450613
55070
59481!
638491
68172
72454
8.476693*
80892
85050j
89170
03250
97293
8.501208
05267
09200
13098
16961
20790
8.524586
88349
82080
85779
89447
43084
11967
11772
11584'
11402
11225'
11054'
10887 ■
10726
10570
10418
10270
10126'
9987
9851
9719'
9590
9465
9343!
9224
9108
8995
8885
8777'
8672'
8570
8470,
8371'
8276 '
8182!
8091 1
8002'
7914
7828
7745
7663
7583
r505
Seomt ID.' Counc
I
7428
7352
7279
7206
7135
7066
6998
6931
6865
6801
6738
6676
6615
6555
6496
6439
6882
6826
6272
6218
6165
6118
6062
11.758079
5089d[
43885
86885,
80044
38309
166771
11.710144;
08708 I
11.697366
91116
84954
78878
11.672886
66975
61144
65390
49711
44105
11.688570
83105
27708
82878
17111
11908
11.606766
01685;
11.596662;
91696
867871
81982
11.577131!
72382;
67685
63038
58440
53890
11.549387
44930
40519'
86151!
818281
27546
11.523807;
19108'
14950
10830
06750
02707
10.000066
68
71 i
78!
75!
80'
10.000082*
85
87?
90!
9s;
95;
10.000098
10.000101; c
^2 5
10.000115! ,
8l'^
24
271
80
10.000133
Wi
89
42
49j
10.000152
66
69i
62
66
69
5
5
f!
ilj
;i5;
,!«'
6
6
6
6
6
6
11.498702!
94738
90800
86902
88089
79210
11.475414
71651
67920
64221
60558
56916
10.000178; «
77«
80
84'
88
91
10.000195
99
10.000208
07
10
10.000218'
22;
26
81
85
89
10.000843
47
88
68
80
88
9.999984 60.
82 591
29 58
27 57'
85 56
82 55
ao;54
9.999918 > 58 ,
15 52
18.51
10 50
07 49
05 48
9.999902; 47
9.999899 1 46
97i45
94 .44
91148
88:42
9.999885 1 41
82.40
79! 89
76 1 88
78.87:
70;8l!
9.999867 185 1
84.84,
6ljl9<
68 8i
54 '81 1
51
9.999848
44
41
tsiu'
84j»
81 24 1
9.999827 21:
90»
16Mi
i«'w
9.999805,17,
01 IK
9.999797 j IS,
»8!H.
90!IS
86lS'
9.99978S;n
78:W
74I 9
69
65
61
9J90767
58
46
44
46
85
Cotiiie D.
Secant
Cotang.
T^.
88 Did.
91
a Deg.
D.
' j Sii»« I
4 670M llf?
5 60640 Sis
6 6»W9^JJ^
7&M74S1; ^^.
10
11
Ifi
GOMC.
TABLE IV^Loo. Sins, xto.
Cotang.
Tang.
D.
Secant D. Codne
9.999786 60
81 I 69
26 68
22i67
17 '66
18 66
08 64
9.999704 ' 63
9.999699 62
94 61
89 60
86 49
80 48
9.999676 47
70 46
66,46
60 44
66 43
60,42
9.999646 41
40 40
36 89
29 88
24 87
19 86
9.999614 85
08.84
03 83
9.999697 82
92 81
86 80
9.999681 29
75,548
70 27
64 I 26
68 26
6324
9.999647 I 28
41 I 22
86; 21
29 120
19
18
708W Sao
7*214 IS?
77666 ^l
80892 "**
12 84198, ^2
13 8.687469' ^^.^
14 907211 "J;
16 98948) ?;;:
16 971621 ^l
17 8.600882 ^V
18 08489; l^
ao 09784i ^JJ5
12828 "J*
M
26
26
27
28
29
90
84
88
6076
6041
6006
4972
18987
21962
26 &824066
SS854 J"*
86776 AQ9Q
80 89680 ^^
82 46428 %Vi
48274 J?g
"^<^ 4682
87 8.668476 .^«
88 62280 VSi
40 67689 1^
41 70898 JJSJ
42 78080 J*S
48 &676761 ..q.
^ 78406 !!^
^ ^IM' 4870
46 88666 !:;?
47 86272 :J:J
48 88868 Z^l
48 8.681488 .our
60 98998 VSit
61 96648 Jf;^
62 99078 Jfjl
68 8.701689 ZVZ
64 04090 4!??
66 8.706677
66 00049
67
68
68
"09049 JJIJ
11607 JJ5i
18962 AAKt
16888 J^
18800 *^^
r©:
11.467181
68678
60006
46461
42946
89460
86001
11.482669
291641
267861
22484
191081
16807
11.412681 '
092791
06062)
02848!
11.899668
96611,
11.898877
90266
871771
841091
810681
78088'
11.875086
72062:
66146
68224;
60820
11J67487
64672
61726
48898
46089
48298
11.840526
87770!
86082!
82811
29607
26920!
11J24249
21696i
189671
16886
18728
11187
11J08662
06002
08467
00927
11.298411
96910
11.298428
90961
884981
86048
88617
81200
8.648084
46691
60268
63817
67836
60828
64291!
8.667727:
711371
74620'
77877;
81208
84614
8.687796'
91061
94288
97492
8.600677:
03889!
8.606978
10094;
18189
162621
19313'
22843;
8.626862:
28340,
81308>
84266
871841
40093
8US42982
46868
48704
61637
64862
67149
8.659928
62689
66483
68160,
70870
73668
8.676289
78900;
81644!
84172
86784
89381 J
8.691963
94629,
97081!
99617i
8.702189;
04646
8.707140!
09618;
12083
14634
16972
19896
Secant i Cotang.
6012
6962
6914
6866
6819
6773
6727
6682
6638
6695
6652
6510
6468
6427
6387
6847
6808
6270
6232
6194
6158
6121
6085
6060
6016
4981
4947
4913
4880
4848
4816
4784
4768
4722
4691
4661
4631
4602
4673
4644
4617
4488
4461
4484
4407
4380
4864
4828
4803
4277
4262
4228
4203
4179
4165
4182
4108
4086
4062
4040
D.
11.456916
58309
49732
46183
42664
89172'
85709'
11.482278
28868
25480:
22123
18792
15486',
11.412206
08949
057171
02508
11.899828!
96161
11.898022
89906!
868111
837381
806871
77657.
11.374648
71660
686921
65744
62816
69907
11.367018
64147,
61296
48463
46648;
42851 !
11.340072
3781 Ij
34667]
31840'
291301
264371
11.828761
21100:
18466:
15828
13216
10619
11.808087
05471
02919
00888
11.297861
95364
11.292860
90882
87917
86466
88028
80604
10.000265
69
74
78
83
87
92
10.000296
10.000801
U
15
20
10.000326
80
85
8l
40 gl
46,8'
60 gl
10.000356 9
60
66
71
76
81
10.000386'
92
97
10.000408
08
14
10.000419
80
86
l^io
47 10
10.000453 iQ
69
65
71
76
82 10
10.000488 ^0
94
10
1O.OOO6OO,J0
Tang.
"Ssi
10
13
1»,10I
10.000625. 0'
81 /"
87i
44
60
67,:
10.000568=
691
76
82;:
89:
96i
COMC. Il>.
24
18
9.999612 17
06
00
9.999493
87
81
9.999476
69
68
66
60
43
9.999487
81
24
18
• 11
04
16
15
14
13
12
11
10
9
8
7
6
6
4
3
2
1
Sine
"87 Deo.
dii
3 Deo.
TABLB IV.— Loo. Suia, nc.
Sine I D.
i 8.718800
II
2
8
4
5
6
7
8
9
10
11
12
18
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
80
81
32
88
84
85
86
37
38
89
40
41
42
48
44
45
46
47
48
49
50
51
52
58
54
55
56
57
58
59
60
21204
23595
25972
28837
80688
83027
8.785354
87667
89969
42259
44536
46802
8.749055
5129'
53528
55747, „^.„
57965' 22i?
4006
3984
3962
3941
3919
8898
I 3877
8857
3836
3816
3796
8776
3756
3737
3717
! 3698
Coiec. I Tang.
11.281200 8.719896
78796': 21806
76405
74028
71663
69312
669731
D. ! Cotang.  Seont iP.
24204
26588
28959
81817
83663
11.264646 8.785996
60151
8.7623371
64511
66675
68828
70970
73101
8.775228
77333
79434
81524
83605
85675
8.787736
89787
91828
93859
95881
97894
8.799897
8.801892
08876
05852
07819
09777
8.811726
18667
15599
17522
19436
21343
8.828240
25180
27011
28884
30749
82607
8.884456
86297
88130
89956
.41774
48585
3661
3642
3624
8606
3588
8570
3553
3535
3518
3501
8484
8467
3451
8434
3418
3402
8386
3370
3354
3339
3323
8308
3293
8278
3263
8249
8234
3219
3205
3191
3177
3163
3149
3135
3122
3108
8095
8082
3069
3056
3043
3030
3017
Coiine
62383
60031
57741
55464
53198
11.250945
48703
46472
44253
42045
39849
11.237663
85489
33325
31172
29030
26899
11.224777
22667
20566
18476
16895
14325
11.212264
10218
08172
06141
04119
02106
11.200103
11.198108
96124
94148
92181
90228
11.188274
86333
84401
82478
80564
78657
11.176760
74870
72989
71116
69251
67393
11.165544
68703
61870
60044
58226
56415
88317
40626
42922
4520
47479
8.749740
51989
54227
56453
58668
60872
8.763065
65246
67417
69578
7172'
73866
8.775995
78114
80222
82820
84408
86486
8.788554
90613
92662
94701
96731
98752
8.800763
02766
04758
06742
0871
10683
8.812641
14589
16529
18461
20384
22298
8.824205
26103
27992
29874
81748
83613
8.885471
87821
89163
40998
42825
44644
4017
8995
8974
8952
8930
3909
8889
8868
8848
3827
3807
8787
3768
3749
3729
8710
3692
8673
3655
3636
8618
8600
3583
3665
3548
8531
8514
3497
8480
8464
8447
8431
3415
3899
3383
3368
8352
8337
3822
8807
3292
8277
3262
8248
3233
3219
3205
3191
8177
8163
8150
3186
3123
3108
8096
8088
3070
3057
3045
8082
Conne
11.280604 10.000596,, 9.999404
78194; 10.000602 :f 9.999898
75796 09;; 91
78412 16' i! 84
71041. 22,; 78
68688 29 it 71
663371 86 J^j ^
11.264004; 10.000648,. J 9.999857
61683: 50 ;^ 50
59874 57 it 48
570781 64 J.^1 86
547931
525211
71
78
12j
12
11.250260 10.000685,., 9J»99815
48011 92 {n 08
45773 99 i^ 01
43547.10.000706;; 9.999294
413321 14 ;; 86
39128' 21 J^ 79
11.236935 10.000728.9 9.999272
84754; 85;; 65
82588' 48 ;; 57
804221 50 ;; 50
28273 58 ;; 42
261341 65 ^3 85
11.224005 10.000778.^ 9.999227
21886 80 ;% 20
88 il 12
95;^ 05
15592 10.000808 il 9.999197
13514 11 [l 89
11.211446 10.000819,. 9.999181
093871 26 , , 74
07338 84 ;; 66
05299 42 f; 58
03269 50 ;% 50
01248 58 jl 42
11.199287 10.000866,, 9.999184
^' 26
18
Secant Cotang.
19778
17680
97235
95242!
93258'.
91283:
74
82
90
98
18,
89317! 10.000906 1^; 9.999094
D.
11.187369
85411
88471
81589
79616
77702
11^75795
78897
72008
70126
68252
66387
11.164529
62679
60887
59002
57175
55856
Tang.
10.000914,. 9.999086
28}* 77
89 **! 61
47J*; Si
66 JT 44
10.000964,^. 9.9990S6
78
81
90
98
10.001007
10.001016
24
88
42
60
69!
COMC
27
]»
10
02
9.998991
9.998984
76
67
58
60
41
60
59 i
58
57 I
56
551
54
53:
52:
,51
50
:49i
47
46
45.
44
43 I
42
41
40
89
S8
86{
551
34(
38
82j
81
30l
29.
28
27
«l
25!
24'
2S
22
21
20.
19'
18
17.
W.
15
14
IS
12'
11
10'
8
7
6
6;
4:
5!
1
86 Deo.
4DBa.
TABLB IV.— Loo. Sim, na
SiiM
COMG.
Tang.
D.
Cotapg. I Secant 'D.
Godne i
1
3
8
4
5
6
7
8
9
10
11
12
IS
14
16
1«
17
18
19
SO
SI
ss
ss
S4
S6
S6
S7
S8
SO
80
81
8S
88
84
85
88
87
88
80
40
41
48
48
44
46
48
47
48
40
60
61
68
68
64
66
68
67
68
60
60
8^8586
46887
47188
48971
60761
62626
64291
8.868049
67801
69546
61288
88014
64788
8^66465
88166
89888
71585
78255
74988
8^76616
78285
79949
81607
88258
84908
8.888542
88174
89801
91421
98086
94848
8.898248
97842
99482
8^1017
02598
04189
8.906788
07297
08858
10404
11949
18488
8.916082
18660
18078
19591
21108
22810
8.984112
25800
27100
28687
80088
81644
8.988015
84481
86942
87898
88850
40298
8005
2992
2980
2987
2956
2948
2981
2919
2908
2898
2884
2878
2881
2850
2889
2828
2817
2808
2795
2788
2778
2768
2752
2742
2781
2721
2711
2700
2890
2880
2870
2660
2851
2841
2881
2622
2812
2808
2598
2584
2575
2586
2558
2547
2588
2529
2520
2512
2508
2494
2486
2477
2488
2460
2452
2448
2485
2427
2419
2411
11.156415'
54618
52817
51029
49249
47475
45709
11.148951
42199
40454
88717
88986
85262
11.188545
81885
80182
28485
26745
25062
11.128885
21715
20051
18898
16742
15097
11^18458
11826,
10199
08579
06965
05857!
11.108754>
02158
00568
11.098988
97404
95881
11.094264
92708
91147
89596
88051
86512
11.084978
88450
81927
80409
78897
77890
11.075888
74891
72900
71418
69982
68456
11.066985
65519
64058
62602
61150
59704
8.844644
46455
48260
50057
51846
53628
55408
8.857171
58982
60686
62483
64173
65906
8.867632
69851
71064
72770
74469
76162
8.877849!
79529!
81202!
82869
84530,'
86185
8.887833
89476
91112
92742
94866
95984
8.897596
99203
8.900803
02398
03987
05570
8.907147
08719
10285
11846
13401
14951
8.916495
18034
19568
21096
22619
24136
8.925649
27156
28658
80155
31647
33134
8.984616
36093
37565
89032
40494
41952
3019
8007
2995
2982
2970
2958
2946
2935
2923
2911
2900
2888
2877
2866
2854
2843
2832
2821
2811
2800
2789
2779
2768
2758
2747
2737
2727
2717
2707
2697
2687
2677
2667
2658
2648
2638
2629
2620
2610
2601
2592
2583
2574
2565
2556
2547
2538
2530
2521
2512
2503
2495
2486
2478
2470
2461
2453
2445
2437
2430
11.155856
58545
51740
49948
481541
46372;
445971
11.142829'
41068
39314,
37567
35827
84094'
10.001059,,,
068!!!
10.001122,,,
140
149
159,
168
11.182368 10.001177
80649,
28936
27230
25531 !
23838
I
11.122151
20471
18798,
17131
15470
1381 5
11.112167
10524
088881
07258'
056341
04016
11.102404
00797!
11.0991971
97602:
96013,
94430
11.092853
91281
89715
88154
86509
85049
11.083505'
81966
80432
78904
77381
75864
11.074351
72844
. 71842
69845
68353
66866
11.065884
63907
62435
60968
59506
58048
187,
196
205
215
224
10.001234
243!
253
262
272
282
10.001292
301
311
321
831
841
10.001351
861
371
881
, 891
401 1
10.001411:
422,j:
s
452;}J
463
10.001473!
484
494
505;
5151
5261
10.001536
547
5581
569
579
590
10.001601
612
623
634
645
656
9.998941 60
982 I 59
923 58
914 57
905 ; 56
896 55
887 54
58
52
51
50
9.998878
869
860
851
841 ; 49
882 48
9.998828 ' 47
818 46
804 45
795 44
785 . 48
776 42
9.998766
757
747
738
728
718
9.998708
699
689
679
669
659
9.998649
639
629
619
609
599
9.998589
678
568
558
548
587
9.998527
516
506
495
485
474
9.998464
458
442
481
421
410
9.998399
388
377
866
855
844
41
40
89
88
87
86
35
84
33
32
81
80
29
28
27
26
25
24
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
8
2
1
D.
Secant Cotang. D.
Tang.
CoMG. D.
Sine
86 Dsa.
u
5 Deg.
TABLB IV.^Loo. Snn, xtc
hine
I Cosoc. i Tang. ; D. i Cotang. Secant D.' Coane t
u
8.940J96
1
41738
2
43174
3
44006
4
46034
6
47456
6
48874
, 7
8.950287
8
51696
' 9
53100
10'
54499
.11'
55S94
112:
57284
13
8.958670
14.
60052
15
61429
16
62801
17
64170
= 18
65534
19
8.966893
20
6S249
21
6y600
.22
70947
23
722S9
24
73628
125
8w974962
'26
7621*3
!27
77619
.28
78941
129
80259
:»0
81573
81
8J82853
32
841S9
33
85491
84
867S9
85
880S3
86
89374
87
8.990660
38
91943
89
93222'
40
94497
41
95768
42
97036
oirt^ 11.059704 8.94l952i
• 58262 A«^«»
43 8J98299
44 99560
45 9.000816
46 ; 02t>69
47 j 03318
48
04563
49 9.005805
07044
08278
09510
10737
11962
9.013182
14400
15613
16824
18031:
19235 '
Cosine I
2394
2387
2379
2871
2363
2355
2348
2340
2332
2325
2317
2310
2302
2295
22S8
2280
2273
2266
2259
2252
2245
2238
2231
2224
221
2210
2203
2197
2190
2183
2177
2170
2163
2157
2150
2144
2138
2131
2125
2119
2112
2106
2100
2094
2088
2082
2076
2070
2064
2058
2052
2046
2040
2034
2029
2023
2017
2012
2006
IT"
56826.
55394
53966
52544
51126
43404
44852
46295
47734
49168
50597
11.049713 8.952021.
48804
46900
45501
44106
42716
53441
54856
56267
57674
59075
11.041330 8.960473
39948
88571
37199
35830
84466
11.033107
31751.
30400
29053
27711
26372
6l$66
63255
64639
66019
67394
8.968766
70133
71496
72855
74209
75560
^ 11.025088 8.976906
23707
22381
21059,
19741=
18427
11.017117
15811,
14509
13211
^ 11917.
10626
8248
9586
80921
82251
83577
8.984899
86217
87532
88842
90149
91451
11.009340 8.992750
0S057
06778
05503'
04232
02964
11.001701
00440,
10.999184
97931;
96682{
95437i
94045
95337
96624
97908
99188
9.00U465
01738
03007
04272
05534
06792
10.994195 9.008047
2421
2413
2405
2397
2390
2882
2374
2366
2358
2351
2344
2335
2329
2321
2314
2307
2300
2293
2286
2279
2271
2265
2257
2251
2244
2237
2230
2223
2217
2210
2204
2197
2191
2184
2178
2171
2165
2158
2152
2146
2140
2134
212'
1 11.058048
i 56596
55146
i 53705
< 52266
50882
49403
11.047979
46569
45144
43738
42826
' 40925
92956;
91722;
904901
89263
88038
09298
10546'
11790;
13031;
14268
10.986818 9.015502
S5600
8488
88176
81969
80765
Secant
16732
17959
19183
20403
21620
Cotang.
17 I
2121
2115
2109
2103
209'
S091
2085
2080
2074
2068
2062
2056
2051
2045
2040
2033
2028
11.089527!
381841
367451
353611
339811
32606:
11.081284!
29867
28504
27145
25791
24440
11.023094
21752
20414
190791
17749'
16428;
11.015101
I 13783
I 12468
11158
I 09851
08549
i 11.007250
05955,
04663
08876
02092
00812
10.999535
10.001656 ^^
667 j^
678 jj.
689 \l
723 [5
10.001734, oi
"^^19
^^'19.
<6S,q:
791 J^i
10.001803, A
814 \l
826 \l
887 J^
861 20
10.001872 '
884^.
908 *J.
920 20
932 20
10.001944 "
956 ;0'
968
20;
.r '
96998
95728
94466
93208
10.991958
I 90702;
89454
I 88210'
I 86969
 85782:
i 10.984498
88268
82041;
808171
79697.
78880
Tang.
9S0 20[
992 20
10.002004 rr
10.002016 ^J
10.002090 *
103 iJi
115 1}
128 i'
153 1 I
10.002165 on
178 2J
203*}
216 2]
2292
10.002242 *
255*1
268 *i
281 2]
294 2}
10.002320 poi
838*^1
846
859
872
8861
Cotec Id.
9.998844 = 6k}
333 59
322 SS
311 57
300 56
2S9 55
277 54
9.098266 53
255 52
243 51
232 50
220 49 ,
209 48 I
9J»98197 47
186 46;
174 45'
163 44
151 43
139 42
9.998129 41
116 40,
104 89
092 . 8S =
080 87
068 M;
9.998056 S5
044, 14 j
032 S3
020 SI.
OOS SI
9.997996 80
9.997984 29
972 ;tt
959. r
947 tf
935 t5,'
922 S4
9.997910 S3
897 S3'
8S5!S1
872;so:
860 is'
847113
9.997885 1 17 1
822'1<>
809!15
797,14
784'l3;
771, i«:
9JM)77S8!ll,
745 10 1
733 9
719
703
69S; 3
9J976S0
637
654
641
6S8
614
84 Deo.
95
Deo.
TABLE IV.— Loo. Btmn, no.
8iae
D.
20485
21682
22825
24016
25208
26886
»jtS7M7
28744
29918
81089'
82257;
83421
35741!
86896,
880481
89197.
40842
0.0414851
42625
48762
44895
46026
47154
9JM8S79
49400
.50519
51685
527491
58859
«UIMN6'
j 56071
57172
! 58271
I 59867
I 60460
9.061551
68724
64806
65885!
669621
9.0680861
69107;
701761
71242
72806,
78866
75480,
76588
77588!
78681)
79676
9.080719!
81759
82797
88882
84864
85894
2000
1995
1989
1984
1978
1978
1967
1962
1957
1951
1946
1941
1986
1980
1925
1920
1915
1910
1905
1899
1895
1889
1884
1879
1875
1870
1865
1860
1855
1850
1845
1841
1886
1881
1827
1822
1817
1818
1808
1804
1799
1794
1790
1786
1781
1777
in2
1768
1768
1759
1755
1750
1746
1742
1788
1788
1729
1725
1721
1717
CMine
D.
Ootec. I Tang.
D.
10.980765
79565!
788681
77175'
75984
747971
78614;
10i»72488
712561
70082:
68911
67748:
66579;
10.965418
64259:
681041
61952
60808.
596581
19J58515'
678761
9.021620
22834
24044
26251,
26465!
27666;
28852
9.080046
81237
82426
83609
84791'
86969
9i)87144,
88316,
39486
55105'
58974!
62846.
19.951721
60600J
4948l{
488651
47261 j
46141>
10.945084
48929
42828
41729
40683;
89640
10J88449
873611
86276:
36194
84115
38088
10.981964
80893
29824
28758
27694
26684
10.926676
24620
23467
22417
21369
20324
10.919281
18241
17203
16168
16186
14106
40661
41813*
42973
9.044130
46284:
46434
47682
48727
Secuit
9^61008
62144
63277
64407
66636
66669
9.067781
68900
60016
61130
62240
63848
9.064463
66666
66666
67762
68846
69938!
9.071027
72113
78197
74278
75366
76432
9.077606
78676,
79644!
80710!
81773!
82833
9.083891
84947;
86000
87050
880981
89144
2023
2017
2011
2006
2000
1995
1990
1986
1979
1974
1969
1964
1968
1958
1948
1943
1938
1933
1928
1923
1918
1913
1908
1903
1898
1693
1889
1884
1879
1874
1870
1866
1860
1856
1861
1846
1842
1837
1833
1828
1824
1819
1816
1810
1806
1802
1797
1793
1789
1784
1780
1776
1772
1767
1763
1769
1765
1751
1747
1743
Cotang.
Secant ID. I Coone
10.97838U
77166
76956
74749
73645
72345
71148
10.969954
I 68763
I 67675
66391
I 66209
I 64031
10.962856
61684
60516
69349
68187
67027
10.955870
64716
63666;
624181
51278!
I 601311
10.948992 10.002720
478661
i 46723
46593!
I 44465
I 43341
10.002386oo
439i22
^^^22
466^2
10.002480 o^
*^^2S
^^^23
53? 23
648 23
10.002661 „^i
589 28
617,23
631 1
ia002645
669
9.997614
601
688
674
661
673
687
701
716
743
758
772
786,
10.942219
41100
39984!
88870
37760
36662
10.936547
34444
33345
32248
81164
30062
10.928973
27887,
268031
26722
24644
23668.
10.922495
, 21424.
I 20856'
I 19290!
I 182271
17167
10.916109
I 160631
I 14000'
12960
I 11902
I 10866
801
10.002816
830
844
869
873
888
10.002902
917
932
947
961
976
10.002991
10.008006
021
036
061
066
10.003081
096
111
126
142
157
10.003172
Cotang. I D. i Tang.
203
218,
284
249
26
26
Cowc. ID.
647 66 I
684 64
9.997520 53
507 52
493 51
480 . 50
466 49
452 48
9.997489 47
425 46
411 i 45
397 : 44
383 43
869 : 42
9.997855 I 41
341 I 40
327 I 89
313 38
299 87
285 , 86
9.997271 j 35
267 i 34
242 '88
228 i 82
214 : 81
199 180
9.997185 29
170 ' 28
156 I 27
141 > 26
157 i 25
112 124
9.997098 28
083 22
068 I 21
053 i 20
039 1 19
024 18
9.997009 17
9.996994 1 16
979 ! 15
964 14
949 ' 13
934 1 12
9.996919 ! 11
904 j 10
889 1 9
874
868
843
9.996828
812
797
782
766
751
Sine
88 Deo.
M
TABLE IV^LoG. Sunn, nc.
Cotec.
Tang.
Cotang.
Secant D. Codne
9.0&5Sik4
1 S69'2'2
87947
85970
89990
9UXIS
92C'24
8
4
5
6
T 9.093"37
8 94047
: 9
10
11
13
95056
96i>6'2
97 ("^S
9S066
18 9.099065
14 9.10CI062
15 010^6
16 0204$
' 17 03037
I 13 040i5
19 9.105010
I 20 05992
21
23
24
25
26
27
i2d
;29
,30
31
: 82
9.:
84
135
36
'87
j83
<89i
40:
421
'48
46
■«i
49
'50
I 51
152
'58
'54
55
56
57
53
59
60
06973
07951
03927
09901
110S73
11S42
12S09
13774
14737
15693
M 16656
17613
13567
19519
20469
21417
►.122362
283U6
24243
25137
26125
27060
L127993
23925
29354
80731
81706
82630
1.183551
34470
85387
86303
87216
88123
1.139037
89944
40350
41754,
42655
48555
1713
17i»9
17"4
1700
1696
1692
1^5$
1654
16S0
1676
1673
16d3
16^)5
1661
1657
1653
1649
1645
1642
1633
1634
1630
1627
1623
1619
1616
1612
1603
1605
1601
1597
1594
1590
1537
1553
1550
1576
1578
1569
1566
1562
1559
1556
1552
1549
1545
1542
1539
1535
1532
1529
1525
1522
1519
1516
1512
1509
15"6
1503
1500
10.914KI6
13073
12053
11030
liHilO
03992
07976
10.906963
05953
04944
03933
02935
01934
10.90«>935
10.399933
93944
97952
96963
95975
ia894990
941*03
93027
92049
91'.»73
90099
10^39127
33153
87191
86226
85263
84302
10.833344
82337
81433
80431
79531
73533
10.877633
76694
75752
74313
73375
72940
10.872007
71075
70146
69219
6S294
67370
10.866449
65530
64613
63697
62734
61872'
10.860963
60056
I 59150
: 58246.
j 57345i
56445
9.089144
90187
91223
92266
9330:
94336'
95367.
9.096395!
97422
93446
99463
9.100437
01504
9.102519
03532
04542
05550
06556
07559
9.103560
09559
10556
11551
12543
13533
9.114521
15507
16491
17472
18452
19429
9.120404
21877
2284S
23317
242S4
25249
9.126211
27172
28130
29037
30041
30994
9.181944
82893
83839
34734
85726
36667
9.187605
38542
89476
40409
41340
42269
9.143196.
44121
45044,
45966
46S85;
478lt3
1738
1735
1731
1727
1722
1719
1715
1711
1707
1703
1699
1695
1691
1637
16S4
1630
1676
1672
1669
1665
1661
1653
1654
1650
1647
1648
1639
1636
1632
1629
1625
1622
1618
1615
1611
1608
1604
16<n
1597
1594
1591
1537
1534
1581
1577
1574
1571
1567
1564
1561
1558
1555
1551
1548
1545
1542
1539
1535
1532
1529
10.910856 10.008249
09813
087721
07784i
06698
05664
04633
265*
280.^
296 .i
i???
10.903605
02578!
01554'
00532
10.899513
98496!
10.89748r
96468,
95458
94450
93444
92441
10.891440
90441
89444
83449
87457!
86467
10.885479
84498
83509;
82528:
81548:
805711
10.879596'
78628
776521
76683;
75716
74751'
10.873789
72828,
71870
70918
69959!
69006
10.868056
67107
66161
65216
64274
63333
10.862895
61458'
60524
595911
58660'
57781
10.856804
55879;
54956
54034
53116
5219'
10.008359 Q«
488 51
10.003454 c^
470;:
486 2^
^2 27
635 $i
10.008551 L,
667 £
583 ii
000 Jli
616 21
63251
10.003649 '
665 5I'
682$;:
698 5^1
716 .
jSlgg
10.008748 „ft
765^1
4 81 Qoi
798 1.
815 H
I0.003S49 Z.
866,5,
88S°I
900^1
917 H
934 »
10.003951 Z:
968^
10.004002 *r
10.004054 Zl\
072»
080 ?»
106^
141 **
"29
10.004159^
i94,;
212;S
229»
24'r»
9.996751 . 60
735 59
720 '58
704 57
688 56
673:65
657 54
9.996641 53
625 53
610 ' 51
594 50
578 49
562 48
9.996546 47
580 4«,
514 45:
498 44
482 43
465 42
9.996449 41 1
438 40;
417 59'
400 3Si
884 37
86S S6:
9.996851 35
335 31!
818 33!
802 32i
285 31 !
269 30
9.996253 29
235 28i
219 27
202 26
185 25,
168 24
9.996151 23 1
134 22i
117 21
100 20
033 19
066 18
9.996049 17
032 10
015 15
9.995998 14
980! 13 j
968 12!
9.996946 ill
928:10
911
894
876
859
9.995841
833
806
788
771
753
Sne
r
i Coaine  D. , gwamt Cotang.  D.
Tang. I Coiec D
82 Dec.
27
8 Deo.
TABLE IV.— Loo. Sihbs, xto.
Sine
D.
Coicc. Tang.
Cotang. Secant D. Cosine
1
2
8
4
6
6
7
8
»
10
11
12
18
14
16
18
17
18
19
20
21
22
28
24
25
26
27
28
20
80
81
82
88
84
86
88
87
88
80
40
41
42
48
44
46
48
47
48
49
60
61
62
68
64
66
5$
67
68
69
80
9.148655
4453
5849
6243
7186
8026
8915
9.149802
9.150686
1569
2451
8330
4208
9.166088
6957
6830
7700
8569
9435
9.160801
1164
2026
2885
8743
4600
9a66454
6807
7159
8008
8856
9702
9.170647
1889
2230
8070
^ 8908
4744
9.176678
6411
7242
8072
8900
9726
9.180651
1874
2196
8016
8884
4651
9.186466
6280
7092
7908
8712
9519
9.190825
1180
1983
2784
8584
4832
Godne
1496
1493
1490
1487
1484
1481
1478
1476
1472
1469
1466
1463
1460
1457
1454
1451
1448
1445
1442
1439
1436
1433
1430
1427
1424
1422
1419
1416
1413
1410
1407
1405
1402
1899
1896
1894
1391
1388
1886
1883
1380
1377
1874
1372
1869
1866
1364
1861
1859
1356
1353
1851
1348
1346
1843
1341
1338
1336
1338
1880
10.856445
5547
4651
3757
2864
1974
1085
10.850198
10.849814
8431
7549
6670
5792
10.844917
4043
3170
2300
1431
0565
10.839699
8836
7975
7116
6257
5400
10.834546
3693
2841
1992
1144
0298
10.829458
8611
7770
6930
6092
5256
10.824422
3589
2758
1928
1100
0274
10.819449
8626
7804
6984
6166
5349
10.814534
3720
2908
2097
1288
0481
10.809675
8870
8067
7266
6466
566 8
Secant
9.147808
8718
9632
9.150544
1454
2363
8269
9.154174
5077
5978
6877
7776
8671
9.159565
9.160457
1347
3128
4008
9.164892
5774
6654
7682
8409
9284
9.170167
1029
1899
2767
3634
4499
9.175362
6224
7084
7942
8799
9655
9.180508
1360
2211
8059
8907
4752
9.185597
6489
7280
8120
8958
9794
9.190629
1462
2294
3124
3953
4780
9.195606
6430
7253
8074
8894
9718
Cotang.
1526
1523
1520
1617
1514
1611
1508
1505
1502
1499
1496
1493
1490
1487
1484
1481
1479
1476
1473
1470
1467
1464
1461
1458
1455
1453
1450
1447
1444
1442
1439
1436
1433
1431
1428
1425
1423
1420
1417
1415
1412
1409
1407
1404
1402
1399
1396
1393
1391
1389
1886
1884
1381
1379
1376
1374
1371
1869
1366
1364
10.852197
1282
0368
10.849456
8546
7687
6781
10.845826
4923
4022
8123
2225
1329
10.840435
10.839543
8653
7764
6877
5992
10.885108
4226
8846
2468
1591
0716
10.004247
265
283
301
319
336
854
10.004372
390
409,
427
4451'
468
10.004481
499
518
536
554
573
10.004591
610
628
647
eee
684
10.829843 10.004703
8971
8101
7283
6366
5501
722
740
759
778
797
10.824688! 10.004816
37761 886
2916 854
2058
1201
0345
10.819492
8640
7789
6941
6093
5248
10.814403
3561
2720
1880
1042
0206
10.809371
8538
7706
6876
6047
5220
10.804394
3570
2747
1926
1106
0287